Beltrami fields exhibit knots and chaos almost surely
aa r X i v : . [ m a t h . SP ] J un BELTRAMI FIELDS EXHIBITKNOTS AND CHAOS ALMOST SURELY
ALBERTO ENCISO, DANIEL PERALTA-SALAS, AND ´ALVARO ROMANIEGA
Abstract.
In this paper we show that, with probability 1, a random Beltrami field exhibitschaotic regions that coexist with invariant tori of complicated topologies. The motivation toconsider this question, which arises in the study of stationary Euler flows in dimension 3, isV.I. Arnold’s 1965 conjecture that a typical Beltrami field exhibits the same complexity asthe restriction to an energy hypersurface of a generic Hamiltonian system with two degreesof freedom. The proof hinges on the obtention of asymptotic bounds for the number of horse-shoes, zeros, and knotted invariant tori and periodic trajectories that a Gaussian randomBeltrami field exhibits, which we obtain through a nontrivial extension of the Nazarov–Sodintheory for Gaussian random monochromatic waves and the application of different tools fromthe theory of dynamical systems, including KAM theory, Melnikov analysis and hyperbolic-ity. Our results hold both in the case of Beltrami fields on R and of high-frequency Beltramifields on the 3-torus. Introduction
Beltrami fields, that is, eigenfunctions of the curl operator satisfyingcurl u = λu (1.1)on R or on the flat torus T for some nonzero constant λ , are a classical family of stationarysolutions to the Euler equation in three dimensions. However, the significance of Beltramifields in the context of ideal fluids in equilibrium was only unveiled by V.I. Arnold in hisinfluential work on stationary Euler flows. Indeed, Arnold’s structure theorem [1, 2] ensuresthat, under suitable technical assumptions, a smooth stationary solution to the 3D Eulerequation is either integrable or a Beltrami field. In the language of fluid mechanics, anintegrable flow is usually called laminar, so complex dynamics (as expected in Lagrangianturbulence) can only appear in a fluid in equilibrium through Beltrami fields. This connectionbetween Lagrangian turbulence and Beltrami fields is so direct that physicists have evencoined the term “Beltramization” to describe the experimentally observed phenomenon thatthe velocity field and its curl (i.e., the vorticity) tend to align in turbulent regions (seee.g. [17, 28]).Motivated by H´enon’s numerical studies of ABC flows [23], which are the easiest examplesof Beltrami fields, Arnold conjectured [1, 2] that Beltrami fields exhibit the same complexityas the restriction to an energy level of a typical mechanical system with two degrees offreedom. To put it differently, a typical Beltrami field should then exhibit chaotic regionscoexisting with a positive measure set of invariant tori of complicated topology.Although specific instances of chaotic ABC flows in the nearly integrable regime have beenknown for a long time [35], the conjecture is wide open. A major step towards the proof ofthis claim was the construction of Beltrami fields on R with periodic orbits and invarianttori (possibly with homoclinic intersections [11] inside) of arbitrary knotted topology [13, 14].In fluid mechanics, these periodic orbits and invariant tori are usually called vortex lines andvortex tubes, respectively, and in fact the existence of vortex lines of any topology had also been conjectured by Arnold in the same papers. These results also hold [16] in the case ofBeltrami fields on T , which, contrary to what happens in the case of R , have finite energy;this is important for applications because R and T are the two main settings in whichmathematical fluid mechanics is studied. The main drawback of the approach we developedto prove these results is that, while we managed to construct structurally stable Beltramifields exhibiting complex behavior, the method of proof provides no information whatsoeverabout to what extent complex behavior is typical for Beltrami fields.Our objective in this paper is to establish Arnold’s view of complexity in Beltrami fields.To do so, the key new tool is a theory of random Beltrami fields, which we develop here inorder to estimate the probability that a Beltrami field exhibits certain complex dynamics.The blueprint for this is the Nazarov–Sodin theory for Gaussian random monochromaticwaves, which yields asymptotic laws for the number of connected nodal components of thewave. Heuristically, the basic idea is that a Beltrami field satisfying (1.1) can be thought ofas a vector-valued monochromatic wave; however, the vector-valued nature of the solutionsand the fact that we aim to control much more sophisticated geometric objects introducesessential new difficulties from the very beginning.1.1. Overview of the Nazarov–Sodin theory for Gaussian random monochromaticwaves.
The Nazarov–Sodin theory [30], whose original motivation was to understand thenodal set of random spherical harmonics of large order [29], provides a very efficient toolto derive asymptotic laws for the distribution of the zero set of smooth Gaussian functionsof several variables. The primary examples are various Gaussian ensembles of large-degreepolynomials on the sphere or on the torus and the restriction to large balls of translation-invariant Gaussian functions on R d . Most useful for our purposes are their asymptotic resultsfor Gaussian random monochromatic waves, which are random solutions to the Helmholtzequation ∆ F + F = 0 (1.2)on R d . We will henceforth restrict ourselves to the case d = 3 for the sake of concreteness.As the Fourier transform of a solution to the Helmholtz equation (1.2) must be supportedon the sphere of radius 1, the way one constructs random monochromatic waves is the fol-lowing [8]. One starts with a real-valued orthonormal basis of the space of square-integrablefunctions on the unit two-dimensional sphere S . Although the choice of basis is immaterial,for concreteness we can think of the basis of spherical harmonics, which we denote by Y lm .Hence Y lm is an eigenfunction of the spherical Laplacian with eigenvalue l ( l + 1), the index l is a non-negative integer and m ranges from − l to l . The degeneracy of the eigenvalue l ( l + 1)is therefore 2 l + 1. To consider a Gaussian random monochromatic wave, one now sets ϕ ( ξ ) := ∞ X l =0 l X m = − l i l a lm Y lm ( ξ ) (1.3a)on the unit sphere | ξ | = 1, ξ ∈ R , where a lm are independent standard Gaussian randomvariables. One then defines F as the Fourier transform of the measure ϕ dσ , where dσ is thearea measure of the unit sphere. This is tantamount to setting F ( x ) := (2 π ) ∞ X l =0 l X m = − l a lm Y lm (cid:18) x | x | (cid:19) J l + ( | x | ) | x | . (1.3b)The central known result concerning the asymptotic distribution of the nodal componentsof Gaussian random monochromatic waves is that, almost surely, the number of connectedcomponents of the nodal set that are contained in a large ball (and even those of any fixed ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 3 compact topology) grows asymptotically like the volume of the ball. More precisely, let usdenote by N F ( R ) (respectively, N F ( R ; [Σ])) the number of connected components of the nodalset F − (0) that are contained in the ball centered at the origin of radius R (respectively,and diffeomorphic to Σ). Here Σ is any smooth, closed, orientable surface Σ ⊂ R . It isobvious from the definition that N F ( R ; [Σ]) only depends on the diffeomorphism class of thesurface, [Σ]. The main result of the theory —which is due to Nazarov and Sodin [30] in thecase of nodal sets of any topology, and to Sarnak and Wigman when the topology of thenodal sets is controlled [32]— can then be stated as follows. Here and in what follows, thesymbol L −−→ a . s . will be used to denote that a certain sequence of random variables convergesboth almost surely and in mean. Morally speaking, this is a law of large numbers for thenumber of connected components associated with the Gaussian field F . Theorem 1.1.
Let F be a monochromatic random wave. Then there are positive constants ν , ν ([Σ]) such that, as R → ∞ , N F ( R ) | B R | L −−→ a . s . ν , N F ( R ; [Σ]) | B R | L −−→ a . s . ν ([Σ]) . Here Σ ⊂ R is any compact surface as above. Gaussian random Beltrami fields on R . Our goal is then to obtain an extensionof the Nazarov–Sodin theory that applies to random Beltrami fields. As we will discuss laterin the Introduction, this is far from trivial because there are essential new difficulties thatmake the analysis of the problem rather involved.The origin of many of these difficulties is strongly geometric. In contrast to the case ofrandom monochromatic waves (or any other scalar Gaussian field), where the main geometricobjects of interest are the components of its nodal set, in the study of random vector fields weaim to understand structures of a much subtler geometric nature. Among these structures,and in increasing order of complexity, one should certainly consider the following:(i)
Zeros , i.e., points where the vector field vanishes.(ii)
Periodic orbits , which can be knotted in complicated ways.(iii)
Invariant tori , that is, surfaces diffeomorphic to a 2-torus that are invariant underthe flow of the field. They can be knotted too.(iv)
Compact chaotic invariant sets , which exhibit horseshoe-type dynamics and have, inparticular, positive topological entropy.Recall that a horseshoe is defined as a compact hyperbolic invariant set on which the time- T flow of u is topologically conjugate to a Bernoulli shift [22], for some T . Consequently, let usdefine the following quantities:(i) N z u ( R ) denotes the number of zeros of u contained in the ball B R .(ii) Given a (possibly knotted) closed curve γ ⊂ R , N o u ( R ; [ γ ]) denotes the number ofperiodic orbits of u contained in B R that are isotopic to γ .(iii) Given a (possibly knotted) torus T ⊂ R , V t u ( R ; [ T ]) is the volume (understood asthe inner measure) of the set of ergodic invariant tori of u that are contained in B R and are isotopic to T . Ergodic means that we consider invariant tori on which theorbits of u are dense.(iv) N h u ( R ) denotes the number of horseshoes of u contained in the ball B R .Clearly, these quantities only depend on the isotopy class of γ and T .It is not hard to believe that these geometric subtleties give rise to a number of analyticdifficulties. One should mention, however, that there also appear other unexpected analytic ALBERTO ENCISO, DANIEL PERALTA-SALAS, AND ´ALVARO ROMANIEGA difficulties whose origin is less obvious. They are related to the fact that it is not clear howto define a random Beltrami field through an analog of (1.3b). This is because the charac-terization of a monochromatic wave as the Fourier transform of a distribution supported ona sphere is the conceptual base of the simple definition (1.3a), which underlies the equivalentbut considerably more awkward expression (1.3b). Heuristically, analytic difficulties stemfrom the fact that there is not such a clean formula in Fourier space for a general Beltramifield. This is because the three components of the Beltrami field (which are monochromaticwaves) are not independent, so the reduction to a Fourier formulation with independent vari-ables is not trivial. We refer the reader to Section 3, where we explain in detail how to defineGaussian random Beltrami fields in a way that is strongly reminiscent of (1.3b). Later in thisIntroduction we shall also informally discuss the aforementioned difficulties and discuss howwe manage to circumvent them using a combination of ideas from PDE, dynamical systemsand probabilityWe can now state our main result for Gaussian random Beltrami fields on R , as definedin Section 3. Let us emphasize that the picture that emerges from this theorem is fullyconsistent with Arnold’s view of complexity in Beltrami fields; with probability 1, we showthat a random Beltrami field is “partially integrable” in that there is a large volume ofinvariant tori, and simultaneously features many compact chaotic invariant sets and periodicorbits of arbitrarily complex topologies. This coexistence of chaos and order is indeed theessential feature of the restriction to an energy hypersurface of a generic Hamiltonian systemwith two degrees of freedom, as Arnold put it. In this direction, Corollary 1.3 below is quiteillustrative. Theorem 1.2.
Let u be a Gaussian random Beltrami field. Then: (i) The topological entropy of u is positive almost surely. In fact, with probability , lim inf R →∞ N h u ( R ) | B R | > ν h . (ii) With probability , the volume of ergodic invariant tori of u isotopic to a givenembedded torus T ⊂ R and the number of periodic orbits of u isotopic to a givenclosed curve γ ⊂ R satisfy the volumetric growth estimate lim inf R →∞ V t u ( R ; [ T ]) | B R | > ν t ([ T ]) , lim inf R →∞ N o u ( R ; [ γ ]) | B R | > ν o ([ γ ]) . The constants ν h , ν t ([ T ]) and ν o ([ γ ]) above are all positive, for any choice of the curve γ andthe torus T . Corollary 1.3.
With probability , a Gaussian random Beltrami field on R exhibits infinitelymany horseshoes coexisting with an infinite volume of ergodic invariant tori of each isotopytype. Moreover, the set of periodic orbits contains all knot types. Remark 1.4.
The result we prove (see Theorem 6.2) is in fact considerably stronger: wedo not only prescribe the topology of the periodic orbits and the invariant tori we count,but also other important dynamical quantities. Specifically, in the case of periodic orbitswe have control over the periods (which we can pick in a certain interval ( T , T )) and themaximal Lyapunov exponents (which we can also pick in an interval (Λ , Λ )). In the caseof the ergodic invariant tori, we can control the associated arithmetic and nondegeneracyconditions. Details are provided in Section 6.Unlike the case of nodal set components considered in the context of the Nazarov–Sodintheory for Gaussian random monochromatic waves, we do not prove exact asymptotics for ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 5 the quantities we study, but only nontrivial lower bounds that hold almost surely. Withoutgetting technicalities at this stage, let us point out that this is related to analytic difficultiesarising fron the fact that we are dealing with quantities that are rather geometrically non-trivial. If one considers a simpler quantity such as the number of zeros of a Gaussian randomBeltrami field, one can obtain an asymptotic distribution law similar to that of the nodalcomponents of a random monochromatic wave, whose corresponding asymptotic constant caneven be computed explicitly:
Theorem 1.5.
With probability , the number of zeros of a Gaussian random Beltrami fieldsatisfies N z u ( R ) | B R | L −−→ a . s . ν z as R → ∞ . The constant is explicitly given by ν z := c z Z R | Q ( z ) | e − e Q ( z )) dz = 0 . . . . , (1.4) where c z := 21 / / [143 √ π ] , and Q, e Q are the following homogeneous polynomials in fivevariables: Q ( z ) := z z + z − z z − z z z − z z + 2 z z z − z z , (1.5) e Q ( z ) := 18965 z + 4211 ( z + z ) + 4213 ( z + z z + z ) . (1.6)1.3. Random Beltrami fields on the torus.
A Beltrami field on the flat 3-torus T :=( R / π Z ) (or, equivalently, on the cube of R of side length 2 π with periodic boundaryconditions) is a vector field on T satisfying the eigenvalue equationcurl v = λv for some real number λ = 0. It is well-known (see e.g. [10]) that the spectrum of the curloperator on the 3-torus consists of the numbers of the form λ = ±| k | for some vector withinteger coefficients k ∈ Z . Restricting our attention to the case of positive eigenvalues for thesake of concreteness, one can therefore label the eigenvalue by a positive integer L such that λ L = L / . The multiplicity of the eigenvalue is given by the cardinality of the correspondingset of spatial frequencies, Z L := { k ∈ Z : | k | = L } . By Legendre’s three-square theorem, Z L is nonempty (and therefore λ L is an eigenvalue ofthe curl operator) if and only if L is not of the form 4 a (8 b + 7) for nonnegative integers a and b .The Beltrami fields corresponding to the eigenvalue λ L must obviously be of the form u L = X k ∈Z L V Lk e ik · x , for some vectors V Lk ∈ C , where V Lk = V L − k to ensure that the Beltrami field is real-valued.Starting from this formula, in Section 7 we define the Gaussian ensemble of random Beltramifields u L of frequency λ L , which we parametrize by L . The natural length scale of the problemis L / .Our objective is to study to what extent the appearance of the various dynamical objectsdescribed above (i.e., horseshoes, zeros, and periodic orbits and ergodic invariant tori ofprescribed topology) is typical in high-frequency Beltrami fields, which corresponds to thelimit L → ∞ . When taking this limit, we shall always assume that the integer L is admissible , ALBERTO ENCISO, DANIEL PERALTA-SALAS, AND ´ALVARO ROMANIEGA by which we mean that it is congruent with 1, 2, 3, 5 or 6 modulo 8. We will see in Section 7(see also [31]) that this number-theoretic condition ensures that the dimension of the spaceof Beltrami fields with eigenvalue λ L tends to infinity as L → ∞ .To state our main result about high-frequency random Beltrami fields in the torus weneed to introduce some notation. In parallel with the previous subsection, for any closedcurve γ and any embedded torus T , let us respectively denote by N z u L , N h u L , N o u L ([ γ ]) and N t u L ([ T ]) the number of zeros, horseshoes, periodic orbits isotopic to γ and ergodic invarianttori isotopic to T of the field u L , as well as the volume (i.e., inner measure) of these tori,which we denote by V t u L ([ T ]). To further control the distribution of these objects, let usdefine the number of approximately equidistributed ergodic invariant tori, N t , e u L ([ T ]), as thelargest integer m for which u L has m ergodic invariant tori isotopic to T that are at a distancegreater than m − / apart from one another. The number of approximately equidistributedhorseshoes N h , e u L , periodic orbits isotopic to a curve N o , e u L ([ γ ]) and zeros N z , e u L are definedanalogously. Note that, again, the asymptotic information that we obtain is perfectly alignedwith Arnold’s view of complex behavior in typical Beltrami fields. Theorem 1.6.
Let us denote by ( u L ) the parametric Gaussian ensemble of random Beltramifields on T , where L ranges over the set of admissible integers. Consider any contractibleclosed curve γ and any contractible embedded torus T in T . Then: (i) With a probability tending to as L → ∞ , the field u L exhibits an arbitrarily largenumber of approximately distributed horseshoes, zeros, periodic orbits isotopic to γ and ergodic invariant tori isotopic to T . More precisely, for any integer m , lim L →∞ P n min (cid:8) N h , e u L , N t , e u L ([ T ]) , N o , e u L ([ γ ]) , N z , e u L (cid:9) > m o = 1 . Furthermore, the probability that the topological entropy of the field grows at leastas L / and that there are infinitely many ergodic invariant tori of u L isotopic to T also tends to : lim L →∞ P (cid:8) N t u L ([ T ]) = ∞ and h top ( u L ) > ν h ∗ L / (cid:9) = 1 . (ii) The expected volume of the ergodic invariant tori of u L isotopic to T is uniformlybounded from below, and the expected number of horseshoes and periodic orbits iso-topic to γ is at least of order L / : lim inf L →∞ min ( E N h u L L / , E N o u L ([ γ ]) L / , E V t u L ([ T ]) ) > ν ∗ ([ γ ] , [ T ]) . In the case of zeros, the asymptotic expectation is explicit, with ν z given by (1.4) : lim L →∞ E N z u L L / = (2 π ) ν z . Here ν h ∗ and ν ∗ ([ γ ] , [ T ]) are positive constants. Remark 1.7.
As in the case of R , the result we prove in Section 7 is actually stronger in thesense that we have control over important dynamical quantities (which now depend stronglyon L ) describing the flow near the above invariant tori and periodic orbits.1.4. Some technical remarks.
In a way, the cornerstone of the Nazarov–Sodin theoryis their very clever (and non-probabilistic) “sandwich estimate”, which relates the number N F ( R ) of connected components of the nodal set of the Gaussian random field F that arecontained in an arbitrarily large ball B R with ergodic averages of the same quantity involving ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 7 the number of components contained in balls of fixed radius. Two ingredients are key toeffectively apply this sandwich estimate. On the one hand, each nodal component cannot betoo small by the Faber–Krahn inequality, which ensures, in dimension 3, that its volume is atleast cλ − if ∆ F + λ F = 0. On the other hand, to control the connected components thatintersect a large ball but are not contained in it, it suffices to employ the Kac–Rice formulato derive bounds for the number of critical points of a certain family of Gaussian randomfunctions.In the setting of random Beltrami fields, the need for new ideas becomes apparent themoment one realizes that there are no reasonable substitutes for these two key ingredients.That is, the frequency λ does not provide bounds for the size of the more sophisticatedgeometric objects considered in this context (i.e., periodic orbits, invariant tori or horseshoes),and one cannot estimate the objects that intersect a ball but are not contained in it using aKac–Rice formula. As a matter of fact, we have not managed to obtain any useful bounds forthese quantities and, while we do use a sandwich inequality of sorts (or at least lower boundsthat can be regarded as a weaker substitute thereof), even the measurability of the variousobjects of interest becomes a nontrivial issue due to their complicated geometric properties.To circumvent these problems, we employ different kinds of techniques. Firstly, ideas fromthe theory of dynamical systems play a substantial role in our proofs. On the one hand,KAM theory and hyperbolic dynamics are important to prove that certain carefully chosenfunctionals are lower semicontinuous, which is key to solve measurability issues that wouldbe very hard to deal with otherwise. Furthermore, to prove that Beltrami fields exhibitchaotic behavior almost surely, it is essential to have at least one example of a Beltrami fieldthat features a horseshoe, and even that was not known. Indeed, the available examples ofnon-integrable ABC flows are known to be chaotic on T due to the non-contractibility ofthe domain, but not on R . This technical point is fundamental, and makes them unsuitablefor the study of random Beltrami fields. Therefore, an important step in our proof is toconstruct, using Melnikov theory, a Beltrami field on R that has a horseshoe. Techniquesfrom Fourier analysis and from the global approximation theory for Beltrami fields are alsonecessary to handle the inherent difficulties that stem from the fact that the equation underconsideration is more complicated than that of a monochromatic wave. As an aside, the onlypoint of the paper where we use the Kac–Rice formula is to compute the constant ν z in closedform.In the case of Beltrami fields on the torus, the results we prove concern not only theexpected values of the quantities of interest, but also the probability of events. In the caseof random monochromatic waves on the torus, Nazarov and Sodin [30] had proved resultsfor the expectation (which apply to very general parametric scalar Gaussian ensembles), andRozenshein [31] had derived very precise exponential bounds for the probability akin to thoseestablished by Nazarov and Sodin [29] for random spherical harmonics. However, both resultsuse in a crucial way that the size of nodal components can be effectively estimated in terms ofthe frequency: the Faber–Krahn inequality provides a lower bound for the volume and largediameter components can be ruled out using a Crofton-type formula and B´ezout’s theorem.No such bounds hold in the case of Beltrami fields, so the way we pass from the informationthat the rescaled covariant kernel of u L tends to that of u to asymptotics for the distributionof invariant tori, horseshoes or periodic orbits is completely different. Specifically, we rely ona direct argument ensuring the weak convergence of sequences of probability measures, onspaces of smooth functions, provided that suitable tightness conditions are satisfied.1.5. Outline of the paper.
In Section 2, we start by describing Beltrami fields in R fromthe point of view of Fourier analysis and provide some results about global approximation. ALBERTO ENCISO, DANIEL PERALTA-SALAS, AND ´ALVARO ROMANIEGA
Gaussian random Beltrami fields on R are introduced in Section 3, where we also establishseveral results about the structure of the corresponding covariance matrix and about theinduced probability measure on the space of smooth vector fields. In Section 4 we recall, in aform that will be useful in later sections, several previous results about ergodic invariant toriand periodic orbits arising in Beltrami fields. Section 5 is devoted to constructing a Beltramifield on R that is stably chaotic. Finally, in Sections 6 and 7 we complete the proofs of ourmain results in the case of R and T , respectively. The paper concludes with an Appendixwhere we provide a fairly complete Fourier-theoretic characterization of Beltrami fields.2. Fourier analysis and approximation of Beltrami fields
In what follows, we will say that a vector field u on R is a Beltrami field ifcurl u = u . Taking the curl of this equation and using that necessarily div u = 0, it is easy to see that u must also satisfy the Helmholtz equation:∆ u + u = 0 . To put it differently, the components of this vector field are monochromatic waves. Animmediate consequence of this is that the Fourier transform b u of a polynomially boundedBeltrami field is a (vector-valued) distribution supported on the unit sphere S := { ξ ∈ R : | ξ | = 1 } . Since u is real-valued, b u must be Hermitian, i.e., b u ( ξ ) = b u ( − ξ ). Furthermore, a classicalresult due to Herglotz [26, Theorem 7.1.28] ensures that if u is a Beltrami field with thesharp fall off at infinity, then there is a Hermitian vector-valued function f ∈ L ( S , C ) suchthat b u = f dσ ; for the benefit of the reader, details on this and other related matters aresummarized in Appendix A. For short, we shall simply write this relation as u = U f , with U f ( x ) := Z S f ( ξ ) e iξ · x dσ ( ξ ) . (2.1)Obviously U f is a Beltrami field if and only if f is Hermitian (which makes U f real valued)and if it satisfies the distributional equation on the sphere iξ × f ( ξ ) = f ( ξ ) . (2.2)In this paper, we are particularly interested in Beltrami fields of the form u = U f , wherenow f is a general Hermitian vector-valued distribution on the sphere. The correspondingintegral, which is convergent if f is integrable, must be understood in the sense of distributionsfor less regular f (that is to say, for f in the scale of Sobolev spaces H s ( S , C ) with s < k > U f is bounded as [15, Appendix A]sup R> R Z B R | U f ( x ) | | x | k dx C k f k H − k ( S , C ) . (2.3)We recall that, for any real s , the H s ( S ) norm of a function f can be computed as k f k H s ( S ) = ∞ X l =0 l X m = − l ( l + 1) s | f lm | , where f lm are the coefficients of the spherical harmonics expansion of f .With q ( t ) := ( π ) / (1 + √ i t ), let us consider the vector-valued polynomial p ( ξ ) := q ( ξ ) ( ξ − , ξ ξ − iξ , ξ ξ + iξ ) , (2.4) ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 9 which we will regard as a Hermitian function p : R → C . Note that the restriction of p to the sphere vanishes exactly at the poles ξ ± := ( ± , , q ( ξ ) has been introduced for later convenience: when we define randomBeltrami fields via the function p in Section 3, this choice of p will ensure that the associatedcovariance matrix is the identity on the diagonal (see Corollary 3.5).We next show that, away from the poles, the density f of a Beltrami field U f must pointin the same direction as p : Proposition 2.1.
The following statements hold: (i)
If the vector field U f is a Beltrami field, then p × f = 0 as a distribution on S . Further-more, if χ is a smooth real-valued function on the sphere supported in S \{ ξ + , ξ − } and f ∈ H s ( S , C ) for some real s , then there is a Hermitian scalar function ϕ ∈ H s ( S ) such that χ f = ϕ p . (ii) Conversely, for any Hermitian ϕ ∈ H s ( S ) , the associated field U ϕp is a Beltramifield.Proof. In view of Equation (2.2), for each vector ξ ∈ S , consider the linear map M ξ on C defined as M ξ V := V − iξ × V .
More explicitly, M ξ is the matrix M ξ = − − iξ iξ iξ − − iξ − iξ iξ − . The determinant of this matrix is det M ξ = ξ + ξ + ξ −
1, and in fact it is easy to see that M ξ has rank 2 for any unit vector ξ . Since M ξ p ( ξ ) = 0 for all ξ ∈ S and p ( ξ ) only vanishesif ξ = ξ ± , we then obtain that the kernel of M ξ is spanned by the vector p ( ξ ) whenever ξ isnot one of the poles ξ ± . In a neighborhood of the poles, the kernel of M ξ can be describedas the linear span of e p ( ξ ) := q ( ξ ) ( ξ ξ + iξ , ξ − , ξ ξ − iξ ).Since M ξ f ( ξ ) = 0 in the sense of distributions by (2.2), it stems from the above analysisthat one can write f ( ξ ) = α ( ξ ) p ( ξ )for ξ away from the poles, and f ( ξ ) = β ( ξ ) e p ( ξ )in a neighborhood of the poles; here α and β are complex-valued scalars. As p ( ξ ) × e p ( ξ ) = 0for all ξ ∈ S , we immediately infer that p × f = 0 . Also, as the support of a function is a closed set, p is bounded away from zero on the supportof χ , so we have that ϕ := χ f · p | p | ∈ H s ( S ) . As f is Hermitian, this proves the first part of the proposition. The second statement followsimmediately from the fact that M ξ [ ϕ ( ξ ) p ( ξ )] = ϕ ( ξ ) M ξ p ( ξ ) = 0 . (cid:3) Remark 2.2.
A Beltrami field of the form U ϕp can be written in terms of the scalar function ψ ( x ) := − R S e iξ · x q ( ξ ) ϕ ( ξ ) dσ ( ξ ) (which satisfies the equation ∆ ψ + ψ = 0) as U ϕp = (curl curl + curl)( ψ, , . Also, it has the sharp decay bound | U ϕp ( x ) | C k ϕ k L ( S ) / (1 + | x | ). Remark 2.3.
Not any Beltrami field of the form U f can be written as U ϕp for some scalarfunction ϕ : an obvious counterexample is given by f ( ξ ) := (0 , , i ) δ ξ + ( ξ ) + (0 , , − i ) δ ξ − ( ξ ) , (2.5)where δ ξ ± is the Dirac measure supported on the pole ξ ± = ( ± , , p ′ is topological. Indeed, as u is divergence-free, we have that ξ · p ′ ( ξ ) = 0, so p ′ must be a tangent complex-valued vector field on S . By the hairy balltheorem, the real part of p ′ must then have at least one zero ξ ∗ . The equation iξ × p ′ ( ξ ) = p ′ ( ξ )implies that the imaginary part of p ′ also vanishes at ξ ∗ , so in fact p ′ ( ξ ∗ ) = 0. This meansthat densities f such as (2.5), where we can take ξ ∗ := ξ + without any loss of generality,cannot be written in the form ϕp ′ .Intuitively speaking, Proposition 2.1 means that any Beltrami field U f whose density f isnot too concentrated on ξ ± can be approximated globally by a field of the form U ϕp . Moreprecisely, one can prove the following: Proposition 2.4.
Consider a Hermitian vector-valued distribution f on S that satisfies thedistributional equation (2.2) , and define ε f,k := inf (cid:8) k Θ f k H − k ( S ) : Θ ∈ C ∞ ( S ) , Θ( ξ + ) = Θ( ξ − ) = 1 (cid:9) . If ε f,k is finite and ε > ε f,k , one can then take a Hermitian scalar distribution on the sphere ϕ , which is in fact a finite linear combination of spherical harmonics if f ∈ H − k ( S , C ) , suchthat sup R> R Z B R | U f ( x ) − U ϕp ( x ) | | x | k dx < Cε . Furthermore, ε f, = 0 if f ∈ L ( S , C ) .Proof. The first assertion is a straightforward consequence of the first part of Proposition 2.1and of the estimate (2.3). Indeed, since f is a compactly supported distribution, then f ∈ H s ( S , C ) for some s . Take any ε ′ ∈ ( ε f,k , ε ) and let us consider a function Θ as above suchthat k Θ f k H − k ( S ) < ε ′ . Since ε ′ > ε f,k , it is obvious that we can assume that Θ = 1 in asmall neighborhood of the poles ξ ± . Applying Proposition 2.1 we infer that χf = ϕp with χ := 1 − Θ and some Hermitian scalar function ϕ ∈ H s ( S ). In view of the fact that the map f U f is linear and of the bound (2.3), we then havesup R> R Z B R | U f ( x ) − U ϕp ( x ) | | x | k dx = sup R> R Z B R | U Θ f ( x ) | | x | k dx C k Θ f k H − k ( S , C ) < Cε ′ . As finite linear combinations of spherical harmonics are dense in H s ( S ), if s = − k we canapproximate ϕ in the H − k ( S ) norm by a Hermitian function ϕ ′ of this form; thensup R> R Z B R | U f ( x ) − U ϕ ′ p ( x ) | | x | k dx sup R> R Z B R | U f ( x ) − U ϕp ( x ) | | x | k dx + sup R> R Z B R | U ( ϕ ′ − ϕ ) p ( x ) | | x | k dx < Cε ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 11 provided that k ϕ − ϕ ′ k H − k ( S ) < ε − ε ′ .Finally, to see that ε f, = 0 if f ∈ L ( S , C ), let us take a smooth function Θ : R → [0 , n ( ξ ) := Θ( nξ − nξ + ) + Θ( nξ − nξ − ) , we trivially get that k Θ n f k L ( S ) k f k L ( S ) for all n > n f tends to zero al-most everywhere in S as n → ∞ . The dominated convergence theorem then shows that k Θ n f k L ( S ) → n → ∞ , thus proving the claim. (cid:3) Another, rather different in spirit, formulation of the principle that densities of the form ϕp can approximate general Beltrami fields is presented in the following theorem. Unlike theprevious corollary, the approximation is considered only locally in space, and in this directionone shows that even considering smooth functions ϕ is enough to obtain a subset of Beltramifields that is dense in the C k compact-open topology: Proposition 2.5.
Fix any positive reals ε and k and a compact set K ⊂ R such that R \ K is connected. Then, given any vector field v satisfying the equation curl v = v inan open neighborhood of K , there exists a Hermitian finite linear combination of sphericalharmonics ϕ such that the Beltrami field U ϕp approximates v in the set K as k U ϕp − v k C k ( K ) < ε . Proof.
Let us fix an open set V ⊃ K whose closure is contained in the open neighborhoodwhere v is defined, and a large ball B R ⊃ V . Since R \ K is connected, it is obvious that wecan take V so that R \ V is connected as well. By the approximation theorem with decay forBeltrami fields [14, Theorem 8.3], there is a Beltrami field w that approximates v as k w − v k C k ( V ) < ε and is bounded as | w ( x ) | < C/ | x | . As the Fourier transform of w is supported on S , Herglotz’stheorem [26, Theorem 7.1.28] shows that one can write w = U f for some vector-valuedHermitian field f ∈ L ( S , C ) that satisfies the distributional equation (2.2). Proposition 2.4then shows that there exists some Hermitian scalar function ϕ ∈ C ∞ ( S ) such that k U f − U ϕp k L ( B R ) < Cε , so that k v − U ϕp k L ( V ) < Cε . As the difference v − U ϕp satisfies the Helmholtz equation∆( v − U ϕp ) + v − U ϕp = 0in V , and K ⊂⊂ V , standard elliptic estimates then allow us to promote this bound to k v − U ϕp k C k ( K ) < Cε , as we wished to prove. (cid:3) Gaussian random Beltrami fields
The Fourier-theoretical characterization of Beltrami fields presented in the previous sectionpaves the way to the definition of random Beltrami fields.In parallel with (1.3a) (see Appendix A for further heuristics), let us start by setting ϕ ( ξ ) := ∞ X l =0 l X m = − l i l a lm Y lm ( ξ ) , where a lm are normally distributed independent standard Gaussian random variables and Y lm is an orthonormal basis of (real-valued) spherical harmonics on S . Note that ϕ is Hermitianbecause of the identity Y lm ( − ξ ) = ( − l Y lm ( ξ ). We now define a Gaussian random Beltramifield as u := U ϕp , where we recall that U f and p were respectively defined in (2.1) and (2.4). Remark 3.1.
As discussed in Proposition 2.1, the role of the vector field p is to ensure thatthe density f := ϕp satisfies the Beltrami equation in Fourier space, iξ × f ( ξ ) = f ( ξ ). Henceone could replace p ( ξ ) by any nonvanishing multiple of it, that is, by e p ( ξ ) := Λ( ξ ) p ( ξ ) whereΛ : R → C is a smooth scalar Hermitian function that does not vanish on S . All the resultsof the paper about random Beltrami fields remain valid if one defines a Gaussian randomBeltrami field as u := U ϕ e p with ϕ as above, provided that one replaces p by e p in the formulas.Also, the results do not change if one replaces the basis of spherical harmonics by any otherorthonormal basis of L ( S ), but this choice leads to slightly more explicit formulas for certainintermediate objects that appear in the proofs.In what follows, we will use the notation D := − i ∇ . An important role will be played bythe vector-valued differential operator with real coefficients p ( D ), whose expression in Fourierspace is \ p ( D ) ψ ( ξ ) = p ( ξ ) b ψ ( ξ ) , for any scalar function ψ in R . Equivalently, by Remark 2.2, the operator p ( D ) reads, inphysical space, as p ( D ) ψ = − (curl curl + curl)( q ( D ) ψ, , , where D := − i∂ x .The first result of this section shows that a Gaussian random Beltrami field is a well definedobject both in Fourier and physical spaces: Proposition 3.2.
With probability , the function ϕ is in H − − δ ( S ) \ L ( S ) for any δ > .In particular, almost surely, u is a C ∞ vector field and can be written as u ( x ) = (2 π ) ∞ X l =0 l X m = − l a lm p ( D ) " Y lm (cid:18) x | x | (cid:19) J l + ( | x | ) | x | / . (3.1) The series converges in C k uniformly on compact sets almost surely, for any k .Proof. For l > − l m l , a lm are independent, identically distributed randomvariables with expected value 1. As the number of these variables with l n is n X l =0 l X m = − l n + 1) , the strong law of large numbers ensures that the sample average, i.e., the random variable X n := 1( n + 1) n X l =0 l X m = − l a lm , converges to 1 almost surely as n → ∞ . Now consider the truncation ϕ n ( ξ ) := n X l =0 l X m = − l i l a lm Y lm ( ξ ) . ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 13
As the spherical harmonics Y lm are orthonormal, the L norm of ϕ n is k ϕ n k L ( S ) = n X l =0 l X m = − l a lm = ( n + 1) X n , and k ϕ n k L ( S ) tends to k ϕ k L ( S ) (which may be infinite) as n → ∞ . Since X n → n + 1) − k ϕ n k L ( S ) tends to 1 almost surely.Therefore, ϕ is not in L ( S ) with probability 1.On the other hand, since k ϕ k H − s ( S ) = ∞ X l =0 l X m = − l a lm ( l + 1) s , it is straightforward to see that the expected value E k ϕ k H − − δ ( S ) = ∞ X l =0 l X m = − l E a lm ( l + 1) δ = ∞ X l =0 l + 1( l + 1) δ is finite for all δ >
0. Hence ϕ ∈ H − − δ ( S ) almost surely, so u := U ϕp is well defined withprobability 1.To prove the representation formula for u and its convergence, let us begin by noting that U i l Y lm p ( x ) = Z S i l p ( ξ ) Y lm ( ξ ) e iξ · x dσ ( ξ )= p ( D ) Z S i l Y lm ( ξ ) e iξ · x dσ ( ξ ) . Using either the theory of point pair invariants and zonal spherical functions [8, Proposition 4]or special function identities [15, Proposition 2.1], the Fourier transform of Y lm dσ has beenshown to be Z S i l Y lm ( ξ ) e iξ · x dσ ( ξ ) = (2 π ) Y lm (cid:18) x | x | (cid:19) J l + ( | x | ) | x | / . This permits to formally write u as (3.1). To show that this series converges in C k on compactsets, for any large n , any N > n and any fixed positive integer k consider the quantity q n,N ( x ) := X | α | k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X l = n l X m = − l a lm D α p ( D ) " Y lm (cid:18) x | x | (cid:19) J l + ( | x | ) | x | / , where we are using the standard multiindex notation. Since p ( D ) is a third-order operator,for all | x | < R we obviously have q n,N ( x ) C k N X l = n l X m = − l | a lm |k Y lm k C k +3 ( S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J l + ( r ) r / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +3 ((0 ,R )) C k N X l = n l X m = − l a lm ( l + 1) δ ! N X l = n l X m = − l ( l + 1) δ k Y lm k C k +3 ( S ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J l + ( r ) r / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +3 ((0 ,R )) where here r := | x | and we have used the Cauchy–Schwartz inequality to pass to the secondline. The Sobolev inequality immediately gives k Y lm k C k +3 ( S ) C k Y lm k H k +5 ( S ) C ( l + 1) k +5 . To estimate the Bessel function, recall the large-degree asymptotics J ν ( r ) ∼ (2 πν ) − (cid:16) er ν (cid:17) ν , which holds as ν → ∞ for uniformly bounded r . As the derivative of a Bessel function canbe written in terms of Bessel functions via the recurrence relation ddr J ν ( r ) = − J ν +1 ( r ) + νr J ν ( r ) , it follows that the C k +3 norm of J l + ( r ) /r / tends to 0 exponentially as l → ∞ on compactsets: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) J l + ( r ) r / (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k +3 ((0 ,R )) (cid:18) CRl (cid:19) l − k − . Since we have proven that ∞ X l =0 l X m = − l a lm ( l + 1) δ < ∞ almost surely, now one only has to put together the estimates above to see that, almost surely, q n,N ( x ) tends to 0 as n → ∞ uniformly for all N > n and for all x in a compact subset of R .This establishes the convergence of the series and completes the proof of the proposition. (cid:3) Remark 3.3.
Note that each summand U i l Y lm p = (2 π ) / p ( D )[ Y lm ( x | x | ) | x | − / J l + ( | x | )] ofthe series (3.1) is a Beltrami field.Since a lm are standard Gaussian variables, it is obvious that the vector-valued Gaussianfield u has zero mean. Our next goal is to compute its covariance kernel, κ , which maps eachpair of points ( x, y ) ∈ R × R to the symmetric 3 × κ ( x, y ) := E [ u ( x ) ⊗ u ( y )] . (3.2)In particular, we show that this kernel is translationally invariant, meaning that it onlydepends on the difference: κ ( x, y ) = κ ( x − y ) . We recall that, by Bochner’s theorem, there exists a nonnegative-definite matrix-valued mea-sure ρ such that κ is the Fourier transform of ρ : this is the spectral measure of the Gaussianrandom field u . In the statement, p j is the j th component of the vector field p . Proposition 3.4.
The components of the covariance kernel of the Gaussian random field u are κ jk ( x, y ) = κ jk ( x − y ) with κ jk ( x ) := (2 π ) p j ( D ) p k ( − D ) J / ( | x | ) | x | / . The spectral measure is dρ ( ξ ) = p ( ξ ) ⊗ p ( ξ ) dσ ( ξ ) . ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 15
Proof. As a lm are independent standard Gaussian variables, E ( a lm a l ′ m ′ ) = δ ll ′ δ mm ′ , so thecovariance matrix is κ jk ( x, y ) = E [ u j ( x ) u k ( y )] = E [ u j ( x ) u k ( y )]= ∞ X l =0 l X m = − l ∞ X l ′ =0 l ′ X m = − l ′ i l − l ′ E ( a lm a l ′ m ′ ) Z S Z S e ix · ξ − iy · η p j ( ξ ) p k ( η ) Y lm ( ξ ) Y l ′ m ′ ( η ) dσ ( ξ ) dσ ( η )= ∞ X l =0 l X m = − l Z S Z S e ix · ξ − iy · η p j ( ξ ) p k ( η ) Y lm ( ξ ) Y lm ( η ) dσ ( ξ ) dσ ( η ) . Here we have used that u and the spherical harmonics Y lm are real-valued. Since Y lm is anorthonormal basis, one has that ∞ X l =0 l X m = − l Z S Z S ψ ( ξ ) φ ( η ) Y lm ( ξ ) Y lm ( η ) dσ ( ξ ) dσ ( η ) = Z S ψ ( ξ ) φ ( ξ ) dσ ( ξ )for any functions ψ, φ ∈ L ( S ). Hence we can get rid of the sums in the above formula andwrite κ jk ( x, y ) = Z S e i ( x − y ) · ξ p j ( ξ ) p k ( ξ ) dσ ( ξ ) , (3.3)which yields the formula for the spectral measure of u . Using now that p is Hermitian (i.e., p ( ξ ) = p ( − ξ )) and a well-known representation formula for the Bessel function J / , theabove integral can be equivalently written as Z S e ix · ξ p j ( ξ ) p k ( ξ ) dσ ( ξ ) = p j ( D ) p k ( − D ) Z S e ix · ξ dσ ( ξ )= (2 π ) p j ( D ) p k ( − D ) J / ( | x | ) | x | / . The proposition then follows. (cid:3)
A straightforward corollary is that the Gaussian random Beltrami field u is normalized sothat its covariance matrix is the identity on the diagonal: Corollary 3.5.
For any x ∈ R , κ ( x, x ) = I .Proof. The formula for the spectral measure computed in Proposition 3.4 implies that κ jk ( x, x ) = Z S p j ( ξ ) p k ( ξ ) dσ ( ξ ) . As p is a polynomial, the computation then boils down to evaluating integrals of the form R S ξ α dσ ( ξ ), where α = ( α , α , α ) is a multiindex and ξ α := ξ α ξ α ξ α . These integrals canbe computed in closed form [18]: Z S ξ α dσ ( ξ ) = ( (cid:2) Q j =1 Γ( α j +12 ) (cid:3) / Γ( | α | +32 ) if α , α , α are even,0 otherwise. (3.4)Here Γ denotes the Gamma function.Armed with this formula and taking into account the explicit expression of the polynomial p ( ξ ) (cf. Equation (2.4)), a tedious but straightforward computation shows Z S p j ( ξ ) p k ( ξ ) dσ ( ξ ) = δ jk . The result then follows. (cid:3)
Remark 3.6.
The probability density function of the Gaussian random vector u ( x ) is there-fore ρ ( y ) := (2 π ) − e − | y | . That is, P { u ( x ) ∈ Ω } = R Ω ρ ( y ) dy for any x ∈ R and any Borelsubset Ω ⊂ R .Since the Gaussian field u is of class C ∞ with probability 1 by Proposition 3.2, it is standardthat it defines a Gaussian probability measure, which we henceforth denote by µ u , on thespace of C k vector fields on R , where k is any fixed positive integer. This space is endowedwith its usual Borel σ -algebra S , which is the minimal σ -algebra containing the “squares” I ( x, a, b ) := { w ∈ C k ( R , R ) : w ( x ) ∈ [ a , b ) × [ a , b ) × [ a , b ) } for all x, a, b, ∈ R . To spell out the details, let us denote by Ω the sample space of the randomvariables a lm and show that the random field u is a measurable map from Ω to C k ( R , R ).Since the σ -algebra of C k ( R , R ) is generated by point evaluations, it suffices to show that u ( x ) = ∞ X l =0 l X m = − l a lm U i l Y lm p ( x )is a measurable function Ω → R for each x ∈ R . But this is obvious because u ( x ) is thelimit of finite linear combinations (with coefficients in R ) of the random variables a lm , whichare of course measurable. In what follows, we will not mention the σ -algebra explicitly tokeep the notation simple. Also, in view of the later applications to invariant tori, we willhenceforth assume that k > κ ( x, y ) only depends on x − y and that the spectral measure has noatoms one can infer two useful properties of our Gaussian probability measure that will beextensively employ in the rest of the paper. Before stating the result, let us recall thatthe probability measure µ u is said to be translationally invariant if µ u ( τ y A ) = µ u ( A ) forall A ⊂ S and all y ∈ R . Here τ y denotes the translation operator on C k fields, defined as τ y w ( x ) := w ( x + y ). Proposition 3.7.
The probability measure µ u is translationally invariant. Furthermore, if Φ is an L random variable on the probability space ( C k ( R , R ) , S , µ u ) , then lim R →∞ − Z B R Φ ◦ τ y dy = E Φ both µ u -almost surely and in L ( C k ( R , R ) , µ u ) .Proof. Since the covariance kernel κ ( x, y ) only depends on x − y , the probability measure µ u is translationally invariant. Also, note that ( y, w ) τ y w defines a continuous map R × C k ( R , R ) → C k ( R , R ) , so the map ( y, w ) Φ( τ y w ) is measurable on the product space R × C k ( R , R ). Wiener’sergodic theorem [30, 5] then ensures that, for Φ as in the statement, there is a random variableΦ ∗ ∈ L ( C k ( R × R ) , µ u ) such that − Z B R Φ ◦ τ y dy L −−→ a . s . Φ ∗ ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 17 as R → ∞ . Furthermore, Φ ∗ is translationally invariant (i.e., Φ ∗ ◦ τ y = Φ ∗ for all y ∈ R almost surely) and E Φ ∗ = E Φ.Also, as the spectral measure (computed in Proposition 3.4 above) has no atoms, a theoremof Grenander, Fomin and Maruyama (see e.g. [30, Appendix B] or [21] and note that theproof carries over to the multivariate and vector-valued case) ensures that the action of thetranslations { τ y : y ∈ R } on the probability space ( C k ( R , R ) , S , µ u ) is ergodic. As themeasurable function Φ ∗ is translationally invariant, one then infers that Φ ∗ is constant µ u -almost surely. As Φ and Φ ∗ have the same expectation, then Φ ∗ = E Φ almost surely. Theproposition then follows. (cid:3)
It is clear that the support of the probability measure µ u must be contained in the spaceof Beltrami fields. In the last result of this section, we show that the support is in fact thewhole space. This property will be key in the following sections. Proposition 3.8.
The support of the Gaussian probability measure µ u is the space of Beltramifields. More precisely, let v be a Beltrami field. For any compact set K ⊂ R and each ε > , µ u (cid:0)(cid:8) w ∈ C k ( R , R ) : k v − w k C k ( K ) < ε (cid:9)(cid:1) > . Proof.
By Proposition 2.5, there exists a Hermitian finite linear combination of sphericalharmonics, ϕ = n X l =0 l X m = − l i l α lm Y lm , where α lm are real numbers (not random variables), such that k v − U ϕp k C k ( K ) < ε/
4. Hence µ u (cid:0)(cid:8) w ∈ C k ( R , R ) : k w − v k C k ( K ) < ε (cid:9)(cid:1) > P (cid:18)(cid:26) k u − U ϕp k C k ( K ) < ε (cid:27)(cid:19) , where P denotes the natural Gaussian probability measure on the space of sequences ( a lm ).Proposition 3.2 shows that the series ∞ X l =0 l X m = − l a lm U i l Y lm p converges in C k ( K ) almost surely, so for any fixed δ > N (whichone can assume larger than n ) such that P (cid:18)(cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = N +1 l X m = − l a lm U i l Y lm p (cid:13)(cid:13)(cid:13)(cid:13) C k ( K ) < ε (cid:27)(cid:19) > − δ . With the convention that α lm := 0 for l > n , note that k u − U ϕp k C k ( K ) N X l =0 l X m = − l | a lm − α lm |k U i l Y lm p k C k ( K ) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = N +1 l X m = − l a lm U i l Y lm p (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) C k ( K ) . Therefore, if we set M := 8( N + 1) max l N max − l m l k U i l Y lm p k C k ( K ) , it follows that P (cid:16)n k u − U ϕp k C k ( K ) < ε o(cid:17) > P (cid:18)(cid:26)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X l = N +1 l X m = − l a lm U i l Y lm p (cid:13)(cid:13)(cid:13)(cid:13) C k ( K ) < ε (cid:27)(cid:19) N Y l =0 l Y m = − l P (cid:16)n | a lm − α lm | < εM o(cid:17) , which is strictly positive. The proposition then follows. (cid:3) Preliminaries about hyperbolic periodic orbits and invariant tori
In this section we construct Beltrami fields that exhibit hyperbolic periodic orbits or apositive measure set of ergodic invariant tori of arbitrary topology. Our constructions arerobust in the sense that these properties hold for any other divergence-free field that is C -close to the Beltrami field. Additionally, we recall some basic notions and results aboutperiodic orbits and invariant tori that will be useful in the following sections.4.1. Hyperbolic periodic orbits.
We recall that a periodic integral curve, or periodicorbit, γ of a vector field u is hyperbolic if all the (possibly complex) eigenvalues λ j of themonodromy matrix of u at γ have modulus | λ j | 6 = 1. Since we are interested in divergence-freevector fields in dimension 3, in this case the eigenvalues are of the form λ, λ − for some real λ >
1. The maximal Lyapunov exponent of the periodic orbit γ is defined as Λ := log λT > T is the period of γ .Given a closed curve γ smoothly embedded in R , we say that γ has the knot type [ γ ]if γ is isotopic to γ . It is well known that the number of knot types is countable. Given aset of four positive numbers I = ( T , T , Λ , Λ ), with 0 < T < T and 0 < Λ < Λ , wedenote by N o u ( R ; [ γ ] , I ) the number of hyperbolic periodic orbits of a vector field u containedin the ball B R , of knot type [ γ ], whose periods and maximal Lyapunov exponents are in theintervals ( T , T ) and (Λ , Λ ), respectively. Since we have fixed the intervals of the periodsand Lyapunov exponents, there is a neighborhood of thickness η of each periodic orbit ( η independent of the orbit) such that no other periodic orbit of this type intersects it. Thecompactness of B R then immediately implies that N o u ( R, [ γ ] , I ) is finite, although the totalnumber of hyperbolic periodic orbits in B R may be countable.An easy application of the hyperbolic permanence theorem [24, Theorem 1.1] implies thatthe above periodic orbits are robust under C -small perturbations, so that N o v ( R ; [ γ ] , I ) > N o u ( R ; [ γ ] , I )for any vector field v that is close enough to u in the C norm. Indeed, if k u − v k C ( B R ) < δ ,then v has a periodic orbit γ δ that is isotopic to, and contained in a tubular neighborhoodof width Cδ of, each periodic orbit γ of u that has the aforementioned properties. Moreover,the period and maximal Lyapunov exponent of γ δ is also δ -close to that of γ , so choosing δ small enough they still lie in the intervals ( T , T ) and (Λ , Λ ), respectively. Thus we haveproved the following: Proposition 4.1.
The functional u N o u ( R ; [ γ ] , I ) is lower semicontinuous in the C k com-pact open topology for vector fields, for any k > . Furthermore, N o u ( R ; [ γ ] , I ) < ∞ for any C vector field u . The following result ensures that, for any fixed knot type [ γ ] and any quadruple I , there isa Beltrami field u for which N o u ( R ; [ γ ] , I ) >
1. This result is a consequence of [13, Theorem1.1], so we just give a short sketch of the proof.
Proposition 4.2.
Given a closed curve γ ⊂ R and a set of numbers I as above, thereexists a Hermitian finite linear combination of spherical harmonics ϕ such that the Beltramifield u := U ϕp has a hyperbolic periodic orbit γ isotopic to γ , whose period and maximalLyapunov exponent lie in the intervals ( T , T ) and (Λ , Λ ) , respectively. ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 19
Proof.
Proceeding as in [13, Section 3, Step 2], after perturbing slightly the curve γ to makeit real analytic (let us also call γ the new curve), we construct a narrow strip Σ that containsthe curve γ . Using the same coordinates ( z, θ ) as introduced in [13, Section 5], we define ananalytic vector field w := | γ | T ∇ θ − Λ z ∇ z , where | γ | is the length of γ and T ∈ ( T , T ), Λ ∈ (Λ , Λ ). Using the Cauchy–Kovalevskayatheorem for Beltrami fields [13, Theorem 3.1], we obtain a Beltrami field v on a neighborhoodof γ such that v | Σ = w . A straightforward computation shows that γ is a hyperbolic periodicorbit of v of period T and maximal Lyapunov exponent Λ. The result immediately followsby applying Proposition 2.5. (cid:3) Corollary 4.3.
There exists R > and δ > such that N o w ( R ; [ γ ] , I ) > for any vectorfield w such that k w − u k C k ( B R ) < δ , provided that k > .Proof. Taking R large enough so that the periodic orbit γ is contained in B R , the result is astraightforward consequence of the lower semicontinuity of N o u ( R ; [ γ ] , I ), cf. Proposition 4.1. (cid:3) Nondegenerate invariant tori.
We recall that an invariant torus T of a vector field u is a compact surface diffeomorphic to the 2-torus, smoothly embedded in R , and such that,the field u is tangent to T and does not vanish on T . In other words, T is invariant underthe flow of u . Given an embedded torus T , we say that T has the knot type [ T ] if T isisotopic to T . It is well known that the number of knot types of embedded tori is countable.To study the robustness of the invariant tori of a vector field it is customary to introducetwo concepts: an arithmetic condition (called Diophantine), which is related to the dynamicsof u on T , and a nondegeneracy condition (called twist) that is related to the dynamics of u in the normal direction to T .We say that the invariant torus T is Diophantine with Diophantine frequency ω if thereexist global coordinates on the torus ( θ , θ ) ∈ ( R / Z ) such that the restriction of the field u to T reads in these coordinates as u | T = a e θ + b e θ , (4.1)for some nonzero real constants a, b , and ω := a/b modulo 1 is a Diophantine number. Thismeans that there exist constants c > ν > (cid:12)(cid:12)(cid:12) ω − pm (cid:12)(cid:12)(cid:12) > cm ν +1 for any integers p, m with m >
1. Here e θ j (often denoted by ∂ θ j ) denotes the tangent vectorin the direction of θ j . We recall that the set of Diophantine numbers (with all c > ν >
1) has full measure. The value of the frequency ω , modulo 1, is independent of thechoice of coordinates.Let us now introduce the notion of twist, which is more involved. To this end, we parame-terize a neighborhood of T with a coordinate system ( ρ, θ , θ ) ∈ ( − δ, δ ) × ( R / Z ) such that T = { ρ = 0 } and u | ρ =0 has the form (4.1). Let us now compute the Poincar´e map π definedby the flow of u on a transverse section Σ ⊂ { θ = 0 } (which exists if δ is small enoughbecause b = 0): π : ( − δ ′ , δ ′ ) × ( R / Z ) → ( − δ, δ ) × ( R / Z ) (4.2)( ρ, θ ) ( π ( ρ, θ ) , π ( ρ, θ )) , (4.3) for δ ′ < δ . Obviously, π (0 , θ ) = (0 , θ + ω ). Since u is divergence-free, the map π preservesan area form σ on Σ, which one can write in these coordinates as σ = F ( ρ, θ ) dρ ∧ dθ , (4.4)for some positive function F . Notice that the area form σ is exact because it can be writtenas σ = dA , where A is the 1-form A := h ( ρ, θ ) dθ , h ( ρ, θ ) := Z ρ − δ F ( s, θ ) ds , and the map π is also exact in the sense that π ∗ A − A is an exact 1-form. Indeed, the areapreservation implies that d ( π ∗ A − A ) = 0; moreover the periodicity of h in θ readily impliesthat Z ( π ∗ A − A ) | ρ =0 = Z ( h (0 , θ + ω ) − h (0 , θ )) dθ = 0 , so the claim follows from De Rham’s theorem. The exactness of both σ and π is a crucialingredient to apply the KAM theory. Remark 4.4.
It was shown in [14, Proposition 7.3] that if the Euclidean volume form dx reads as H ( ρ, θ , θ ) dρ ∧ dθ ∧ dθ in coordinates ( ρ, θ , θ ) for some positive function H ,then the factor F that defines the area form σ is F ( ρ, θ ) = H ( ρ, θ , u θ ( ρ, θ , u θ denotes the θ -component of the vector field u .The twist of the invariant torus T is then defined as the number τ := Z ∂ ρ π (0 , θ ) F (0 , θ ) dθ . (4.5)The reason for which we consider this quantity is that it crucially appears in the KAMnondegeneracy condition of [20], cf. Ref. [14, Definition 7.5] for this particular case.In the present paper we are interested in the volume of the set of invariant tori of adivergence-free vector field u . More precisely, given a quadruple J := ( ω , ω , τ , τ ), where0 < ω < ω , 0 < τ < τ , we denote by V t u ( R ; [ T ] , J ) the inner measure of the set ofDiophantine invariant tori of a vector field u contained in the ball B R , of knot type [ T ],whose frequencies and twists are in the intervals ( ω , ω ) and ( τ , τ ), respectively. One mustemploy the inner measure of this set (as opposed to its usual volume) because this set doesnot need to be measurable. When we speak of the volume of this set, it should always beunderstood in this sense. An efficient way of providing a lower bound for this volume is byconsidering, for each V >
0, the number N t u ( R ; [ T ] , J , V ) of pairwise disjoint (closed) solidtori contained in B R whose boundaries are Diophantine invariant tori with parameters in J and which contain a set of Diophantine invariant tori with parameters in J of inner measuregreater that V . Remark 4.5.
The twist defined in Equation (4.5) depends on several choices we made toconstruct the Poincar´e map (i.e., the transverse section and the coordinate system). Accord-ingly, the functional V t u ( R ; [ T ] , J ) has to be understood as the inner measure of the set ofDiophantine invariant tori whose twists lie in the interval ( τ , τ ) for some choice of (suitablybounded) coordinates and sections, and similarly with N t u ( R ; [ T ] , J , V ). It is well knownthat the property of nonzero twist is independent of the aforementioned choices.Since the Poincar´e map π that we introduced above is exact, we can apply the KAMtheorem for divergence-free vector fields [27, Theorem 3.2] to show that the above invarianttori are robust for C -small perturbations, so that V t v ( R ; [ T ] , J ) > V t u ( R ; [ T ] , J ) + o (1) and N t v ( R ; [ T ] , J , V ) > N t u ( R ; [ T ] , J , V ) for any divergence-free vector field v that is C -close ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 21 to u . Indeed, if k u − v k C ( B R ) < δ , then v has a set of Diophantine invariant tori of knottype [ T ] and of volume V t v ( R ; [ T ] , J ) > V t u ( R ; [ T ] , J ) − Cδ / . Here we have used that the frequency and twist of each of these invariant tori is δ -closeto those of u , so by choosing δ small enough they lie in the intervals ( ω , ω ) and ( τ , τ ),respectively. The argument for N t u ( R ; [ T ] , J , V ) is analogous. Summing up, we have provedthe following: Proposition 4.6.
The functionals u N t u ( R ; [ T ] , J , V ) and u V t u ( R ; [ T ] , J ) are lowersemicontinuous in the C k compact open topology for divergence-free vector fields, for any k > . We next show that, for any knot type [ T ], one can pick a quadruple J and some V > u with N t u ( R ; [ T ] , J , V ) >
1. This is a straightforwardconsequence of [14, Theorem 1.1] (see also [11, Section 3]), so we just sketch the proof.
Proposition 4.7.
Given an embedded torus
T ⊂ R , there exists a set of numbers J , V as above, and a Hermitian finite linear combination of spherical harmonics ϕ such that theBeltrami field u := U ϕp has a set of inner measure greater than V > that consists ofDiophantine invariant tori of knot type [ T ] whose frequencies and twists lie in the intervals ( ω , ω ) and ( τ , τ ) , respectively.Proof. It follows from [14, Theorem 1.1] that there exists a Beltrami field v that satisfiescurl v = λv in R for some small constant λ >
0, which has a positive measure set of invarianttori of knot type [ T ]. These tori are Diophantine and have positive twist. It is obvious that thefield u ( x ) := v ( x/λ ) satisfies the equation curl u = u in R , and still has a set of Diophantineinvariant tori of knot type [ T ] of measure bigger than some constant V , and positive twist.The result follows taking the intervals ( ω , ω ) and ( τ , τ ) in the definition of J , so thatthey contain the frequencies and twists of these tori of u , and applying Proposition 2.5 toapproximate u by a Beltrami field U ϕp in a large ball containing the aforementioned set ofinvariant tori. (cid:3) Corollary 4.8.
Take J and V as in Proposition 4.7. There exists R > and δ > suchthat N t w ( R ; [ T ] , J , V ) > and V t w ( R ; [ T ] , J ) > V / for any divergence-free vector field w such that k w − u k C k ( B R ) < δ , provided that k > .Proof. Taking R large enough so that the aforementioned set of invariant tori of u is con-tained in B R , the result is a straightforward consequence of the lower semicontinuity of N t u ( R ; [ T ] , J , V ) and V t u ( R ; [ T ] , J ), cf. Proposition 4.6. (cid:3) A Beltrami field on R that is stably chaotic Our objective in this section is to construct a Beltrami field u in R that exhibits ahorseshoe, that is, a compact (normally) hyperbolic invariant set on which the time- T flow of u (or of a suitable reparametrization thereof) is topologically conjugate to a Bernoulli shift.It is standard that a horseshoe of a three-dimensional flow is a connected branched surface,and that the existence of a horseshoe is stable in the sense that any other field that is C -closeto u has a horseshoe too [22, Theorem 5.1.2]. Moreover, the existence of a horseshoe impliesthat the field has positive topological entropy; recall that the topological entropy of the field,which we denote as h top ( u ), is defined as the entropy of its time-1 flow. Summarizing, we have the following result for the number of (pairwise disjoint) horseshoes of u contained in B R , N h u ( R ): Proposition 5.1.
The functional u N h u ( R ) is lower semicontinuous in the C k compactopen topology for vector fields, for any k > . Moreover, if u has a horseshoe, its topologicalentropy is positive. In short, the basic idea to construct a Beltrami field with a horseshoe, is to constructfirst “an integrable” Beltrami field having a heteroclinic cycle between two hyperbolic pe-riodic orbits, which we subsequently perturb within the Beltrami class to produce a trans-verse heteroclinic intersection. By the Birkhoff–Smale theorem, this ensures the existence ofhorseshoe-type dynamics.
Proposition 5.2.
There exists a Hermitian finite linear combination of spherical harmon-ics ϕ such that the Beltrami field u := U ϕp exhibits a horseshoe. In other words, N h u ( R ) > for all large enough R > .Proof. Let us take cylindrical coordinates ( z, r, θ ) ∈ R × R + × T , with T := R / π Z , definedas z := x , ( r cos θ, r sin θ ) := ( x , x ) . We now consider the axisymmetric vector field v in R given by v := 1 r (cid:16) ∂ r ψ E z − ∂ z ψ E r + ψr E θ (cid:17) . (5.1)Here ψ := cos z + 3 rJ ( r )with J being the Bessel function of the first kind and order 1, and the vector fields E z := (0 , , , E r := 1 r ( x , x , , E θ := ( − x , x , , which are often denoted by ∂ z , ∂ r , ∂ θ in the dynamical systems literature, have been chosenso that E z · ∇ φ = ∂ z φ , E r · ∇ φ = ∂ r φ , E θ · ∇ φ = ∂ θ φ for any function φ . Notice that v · ∇ ψ = 0, so the scalar function ψ is a first integral of v .This means that the trajectories of the field v are tangent to the level sets of ψ .The vector field v is not defined on the z -axis, so we shall consider the domain in Euclidean3-space Ω := { ( z, r, θ ) : ( z, r ) ∈ D , θ ∈ T } , where D is the domain in the ( z, r )-plane given by D := (cid:26) ( z, r ) : − < z < , < r < (cid:27) . The reason for choosing this particular domain of R will become clear later in the proof; forthe time being, let us just note that ψ ( z, r ) > z, r ) ∈ D .Also, observe that, away from the axis r = 0, the vector field v is smooth and satisfies theBeltrami field equation curl v = v .We claim that, in Ω, v has two hyperbolic periodic orbits joined by a heteroclinic cycle.Indeed, noticing that ( ∂ z ψ, ∂ r ψ ) = ( − sin z, rJ ( r )) , ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 23 where we have used the identity ∂ r [ rJ ( r )] = rJ ( r ), it follows that the points p ± :=( ± π, j , ) ∈ D are critical points of ψ . Here j , = 2 . . . . is the first zero of the Besselfunction J . Plugging this fact in Equation (5.1), this implies that, on the circles in 3-space γ ± := { ( z, r, θ ) : ( z, r ) = p ± , θ ∈ T } , the field v takes the form v ( p ± , θ ) = c j , E θ with c := 3 j , J ( j , ) − >
0. Therefore, we conclude that the circles γ ± are periodic orbitsof v contained in Ω.It is standard that the stability of these periodic orbits can be analyzed using the associatednormal variational equation. Denoting by ( v z , v r , v θ ) the components of the field v in the basis { E z , E r , E θ } , this is the linear ODE ˙ η = Aη , where η takes values in R and A is the constant matrix A := ∂ ( v z , v r ) ∂ ( z, r ) (cid:12)(cid:12)(cid:12)(cid:12) ( z,r )= p ± = (cid:18) J ′ ( j , ) − /j , (cid:19) . The Lyapunov exponents of the periodic orbit γ ± are the eigenvalues of the matrix A . There-fore, since J ′ ( j , ) <
0, these periodic orbits have a positive and a negative Lyapunov expo-nent, so they are hyperbolic periodic orbits of saddle type.Since ψ is a first integral of v and ψ ( p ± ) = c , the set { ( z, r, θ ) : ψ ( z, r ) = c } is an invariant singular surface of the vector field v . This set contains two regular surfaces Γ and Γ diffeomorphic to a cylinder. We label them so Γ is contained in the half space { r j , } and Γ in { r > j , } . The boundaries of these cylinders are the periodic orbits γ ± . Thesurface Γ is the stable manifold of γ + that coincides with an unstable manifold of γ − , whileΓ is the unstable manifold of γ + that coincides with a stable manifold of γ − . Hence theunion Γ ∪ Γ of both cylinders then form an heteroclinic cycle between the periodic orbits γ + and γ − , and one can see that it is contained in Ω.Let us now perturb the Beltrami field v in Ω by adding a vector field w (to be fixed later)that also satisfies the Beltrami field equation curl w = w . Our goal is to break the heterocliniccycle Γ ∪ Γ in order to produce transverse intersections of the stable and unstable manifoldsof γ ε + and γ ε − , where γ ε ± denote the hyperbolic periodic orbits of the perturbed vector field X := v + εw = (cid:18) ∂ r ψr + εw z (cid:19) E z + (cid:18) − ∂ z ψr + εw r (cid:19) E r + (cid:18) ψr + εw θ (cid:19) E θ . As before, ( w z , w r , w θ ) denote the components of the vector field w in the basis { E z , E r , E θ } ,which are functions of all three cylindrical coordinates ( z, r, θ ). If ε > θ -component of X is positive on the domain Ω, so we can divide X by the factor X θ := ψr + εw θ > Y that has the same integral curves up to areparametrization: Y := XX θ = r∂ r ψ + εr w z ψ + εr w θ E z + − r∂ z ψ + εr w r ψ + εr w θ E r + E θ . (5.2)Substituting the expression of ψ ( z, r ) and expanding in the small parameter ε , the analysisof the integral curves of Y reduces to that of the following non-autonomous system of ODEs in the planar domain D : dzdt = 3 r J ( r ) ψ ( z, r ) + ε (cid:18) r w z ( z, r, t ) ψ ( z, r ) − r J ( r ) w θ ( z, r, t ) ψ ( z, r ) (cid:19) + O ( ε ) , (5.3) drdt = r sin zψ ( z, r ) + ε (cid:18) r w r ( z, r, t ) ψ ( z, r ) − r sin z w θ ( z, r, t ) ψ ( z, r ) (cid:19) + O ( ε ) . (5.4)Notice that the dependence on t is 2 π -periodic, and that we have replaced θ by t in thefunction w z ( z, r, θ ) (and similarly w r , w θ ) because the θ -component of the vector field Y is 1.When ε = 0, one has ˙ z = 3 r J ( r ) ψ ( z, r ) , (5.5)˙ r = r sin zψ ( z, r ) . (5.6)Hence the unperturbed system is Hamiltonian with symplectic form ω := r − dz ∧ dr andHamiltonian function H ( z, r ) := log ψ ( z, r ). The periodic orbits γ ± of v and their heterocliniccycle Γ ∪ Γ correspond to the (hyperbolic) fixed points p ± of the unperturbed system joinedby two heteroclinic connections e Γ k := Γ k ∩ { θ = 0 } , k = 1 ,
2. These are precisely the twopieces of the level curve { H ( z, r ) = log c } that are contained in D . Let us denote by γ k ( t ) = ( Z k ( t ; 0 , r k ) , R k ( t ; 0 , r k ))the integral curves of the separatrices that solve Equations (5.5)-(5.6) with initial conditions(0 , r k ) ∈ e Γ k . Of course, the closure of the set { γ k ( t ) : t ∈ R } is e Γ k , and the stability analysisof the periodic integral curves γ ± readily implies that lim t →± ( − k +1 ∞ γ k ( t ) = p ± .By the implicit function theorem, the perturbed system (5.3)-(5.4) has exactly two hy-perbolic fixed points p ε ± ∈ D so that p ε ± → p ± as ε →
0. The technical tool to prove thatthe unstable (resp. stable) manifold of p ε + and the stable (resp. unstable) manifold of p ε − intersect transversely when ε > Y , Y , respectively, as the unperturbed system and the first order in ε perturbation,i.e., Y := 3 r J ( r ) ψ ( z, r ) E z + r sin zψ ( z, r ) E r ,Y := (cid:18) r w z ψ ( z, r ) − r J ( r ) w θ ψ ( z, r ) (cid:19) E z + (cid:18) r w r ψ ( z, r ) − r sin zw θ ψ ( z, r ) (cid:19) E r . Since the unperturbed system is Hamiltonian, we can apply Lemma 5.4 below (which is avariation on known results in Melnikov theory) to conclude that if the Melnikov functions M k ( t ) := Z ∞−∞ ω ( Y , Y ) | γ k ( t − t ) dt , (5.7)have simple zeros for each k = 1 ,
2, then the aforementioned transverse intersections exist, andthat actually the heteroclinic connections intersect at infinitely many points. The integrand ω ( Y , Y ) denotes the action of the symplectic 2-form ω on the vector fields Y , Y , evaluatedon the integral curve γ k ( t − t ). It is standard that the improper integral in the definitionof the Melnikov functions is absolutely convergent because of the hyperbolicity of the fixedpoints joined by the separatrices (see e.g. [22, Section 4.5]). Also notice that although [22,Section 4.5] concerns transverse intersections of homoclinic connections, the analysis appliesverbatim to transverse intersections of heteroclinic connections. ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 25
More explicitly, the Melnikov functions are given by M k ( t ) = 1 c Z ∞−∞ R k ( t ) (cid:2) w z ( Z k ( t ) , R k ( t ) , t ) sin Z k ( t ) − R k ( t ) J ( R k ( t )) w r ( Z k ( t ) , R k ( t ) , t ) (cid:3) dt , where R k ( t ) ≡ R k ( t ; 0 , r k ) and Z k ( t ) ≡ Z k ( t ; 0 , r k ). It is well known that the existence oftransverse intersections is independent of the choice of initial condition.To analyze these Melnikov integrals, let us now choose the particular perturbation w = J ( r ) sin θ E z + J ( r ) r cos θ E r − J ′ ( r ) sin θr E θ . (5.8)It is easy to check that curl w = w in R ; in fact w = (curl curl + curl)( J ( r ) , ,
0) (or, toput it differently, w = U ϕ ′ q ( ξ ) − p , where the distribution ϕ ′ on the sphere S is the Lebesguemeasure of the equator, normalized to unit mass). With this choice, the Melnikov functionstake the form c M k ( t ) = Z ∞−∞ R k ( t ) (cid:2) J ( R k ( t )) sin Z k ( t ) sin( t + t ) − J ( R k ( t )) J ( R k ( t )) cos( t + t ) (cid:3) dt =: a k sin t + b k cos t , where the constants a k , b k are given by the integrals a k = Z ∞−∞ R k ( t ) (cid:2) J ( R k ( t )) sin Z k ( t ) cos t + 3 J ( R k ( t )) J ( R k ( t )) sin t (cid:3) dt ,b k = Z ∞−∞ R k ( t ) (cid:2) J ( R k ( t )) sin Z k ( t ) sin t − J ( R k ( t )) J ( R k ( t )) cos t (cid:3) dt . Since the Hamiltonian function has the symmetry H ( − z, r ) = H ( z, r ), it follows that R k ( t ) = R k ( − t ) and Z k ( t ) = − Z k ( − t ). This immediately yields that a = a = 0. Moreover, it is nothard to compute the constants b and b numerically: b = 3 . . . . , b = 0 . . . . Therefore, the function M k ( t ) = b k cos t is a nonzero multiple of the cosine, so it obviouslyhas exactly two zeros in the interval [0 , π ), which are nondegenerate. It then follows fromLemma 5.4 below that the two heteroclinic connections joining p ε ± intersect transversely.In turn, this implies [33, Theorem 26.1.3] that each hyperbolic fixed point p ε ± has transversehomoclinic intersections, so by the Birkhoff–Smale theorem [22, Theorem 5.3.5] the perturbedsystem (5.3)-(5.4) (with w given by Equation (5.8)) has a compact hyperbolic invariant seton which the dynamics is topologically conjugate to a Bernoulli shift. This set is contained ina neighborhood of the heteroclinic cycle e Γ ∪ e Γ , and hence in the planar domain D where thesystem is defined. This immediately implies that the vector field Y defined in Equation (5.2),which is the suspension of the non-autonomous planar system (5.3), has a compact normallyhyperbolic invariant set K on which its time- T flow is topologically conjugate to a Bernoullishift, where T := 2 πN for some positive integer N >
0. The invariant set K is containedin Ω because it lies in a small neighborhood of the invariant set Γ ∪ Γ . Since the integralcurves of X and Y are the same, up to a reparametrization, K is also a chaotic invariant setof the Beltrami field X in Ω.Finally, since R \ Ω is connected, and of course the vector field X satisfies the Beltramiequation in an open neighborhood of Ω, for each δ >
0, Proposition 2.5 shows that there isa Hermitian finite linear combination of spherical harmonics ϕ such that k X − U ϕp k C (Ω) < δ . If δ is small enough, the stability of transverse intersections implies that the Beltrami field U ϕp has a compact chaotic invariant set K δ in a small neighborhood of K on which a suitablereparametrization of its time- T flow is conjugate to a Bernoulli shift, so the propositionfollows. (cid:3) Corollary 5.3.
There exists R > and δ > such that N h w ( R ) > for any vector field w such that k w − u k C k ( B R ) < δ , provided that k > .Proof. Taking R so that the horseshoe of u is contained in B R , the result is a straightfor-ward consequence of the lower semicontinuity of N h u ( R ), cf. Proposition 5.1. (cid:3) To conclude, the following lemma gives the formula for the Melnikov function that weemployed in the proof of Proposition 5.2 above. This is an expression for the Melnikovfunction of perturbations of a planar system that is Hamiltonian with respect to an arbitrarysymplectic form. This is a minor generalization of the well-known formulas [22, Theorem4.5.3] and [25, Equation (23)], which assume that the symplectic form is the standard one.
Lemma 5.4.
Let Y be a smooth Hamiltonian vector field defined on a domain D ⊂ R withHamiltonian function H and symplectic form ω . Assume that this system has two hyperbolicfixed points p ± joined by a heteroclinic connection e Γ . Take a smooth non-autonomous planarfield Y , which we assume π -periodic in time, and consider the perturbed system Y + εY + O ( ε ) . Then the simple zeros of the Melnikov function M ( t ) := Z ∞−∞ ω ( Y , Y ) | γ ( t − t ; p ) dt , where the integrand is evaluated at the integral curve γ ( t − t ; p ) of Y parametrizing theseparatrix e Γ , give rise to a transverse heteroclinic intersection of the perturbed system, forany small enough ε .Proof. If ε is small enough, the perturbed system has two hyperbolic fixed points p ε ± . Toanalyze how the heteroclinic connection is perturbed, we take a point p ∈ e Γ and we computethe so-called displacement (or distance) function ∆( t ) on a section Σ based at p and trans-verse to e Γ. Recall that the function ε ∆( t ) gives the distance of the splitting, up to order O ( ε ), between the corresponding stable and unstable manifolds of the perturbed system atthe section Σ.A standard analysis, cf. [25, Equation (22)] or the proof of [22, Theorem 4.5.3], yields thefollowing formula for ∆( t ):∆( t ) = 1 | Y ( p ) | Z ∞−∞ Y ( γ ( t − t )) × Y ( γ ( t − t )) e − R t − t Tr DY ( γ ( s )) ds dt , (5.9)where we have omitted the dependence of the integral curve on the initial condition p ∈ e Γ.Here we are using the notation X × Y := X Y − X Y for vectors X, Y ∈ R and Tr DY isthe trace of the Jacobian matrix of the unperturbed field Y .Take coordinates in D , which we will call ( z, r ) just as in the proof of Proposition 5.2, andwrite the symplectic form as ω = ρ ( z, r ) dz ∧ dr , where ρ ( z, r ) is a smooth function that doesnot vanish. Let us call here { e z , e r } the basis of vector fields dual to { dz, dr } (which areusually denoted by ∂ z and ∂ r , as they correspond to the partial derivatives with respect tothe coordinates z and r ). The Hamiltonian field Y reads in these coordinates as Y = 1 ρ ( z, r ) (cid:16) ∂ r H e z − ∂ z H e r (cid:17) . ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 27
Noting that Y ( γ ( t − t )) × Y ( γ ( t − t )) = ω ( Y , Y ) | γ ( t − t ) ρ ( γ ( t − t ))and e − R t − t Tr DY ( γ ( s )) ds = e R t − t Y ( γ ( s )) ·∇ log ρ ( γ ( s )) ds (5.10)= e R t − t d log ρ ( γ ( s )) ds ds = ρ ( γ ( t − t )) ρ ( p ) , (5.11)Equation (5.9) implies that ∆( t ) = M ( t ) | Y ( p ) | ρ ( p ) , so the claim follows because M ( t ) coincides with the displacement function up to a constantproportionality factor. (cid:3) Asymptotics for random Beltrami fields on R We are now ready to prove our main results about random Beltrami fields on R , Theo-rems 1.2 and 1.5. To do this, as we saw in the two previous sections, we need to handle setsthat have a rather geometrically complicated structure, which gives rise to several measura-bility issues. For this reason, we start this section by proving a version of the Nazarov–Sodinsandwich estimate [30, Lemma 1] that circumvents some of these issues and which is suitablefor our purposes.6.1. A sandwich estimate for sets of points and for arbitrary closed sets.
For anysubset Γ ⊂ R , we denote by N ( x, r ; Γ) the number of connected components of Γ that arecontained in the ball B r ( x ). Also, if X := { x j : j ∈ J } , where x j ∈ R , is a countable set ofpoints (which is not necessarily a closed subset of R ), then we define N ( x, r ; X ) := X ∩ B r ( x )]as the number of points of X contained in the open ball B r ( x ). For the ease of notation, wewill write N ( r ; Γ) := N (0 , r ; Γ) and similarly N ( r ; X ). We remark that these numbers maybe infinite. Lemma 6.1.
Let Γ be any subset of R whose connected components are all closed and let X := { x j : j ∈ J } , with x j ∈ R , be a countable set of points of R . Then the functions N ( · , r ; X ) and N ( · , r ; Γ) are measurable, and for any < r < R one has Z B R − r N ( y, r ; X ) | B r | dy N ( R ; X ) Z B R + r N ( y, r ; X ) | B r | dy , Z B R − r N ( y, r ; Γ) | B r | dy N ( R ; Γ) . Proof.
Let us start by noticing that N ( y, r ; X ) = { j ∈ J : x j ∈ B ( y, r ) } = X j ∈J B r ( x j ) ( y ) . As the ball B r ( x ) is an open set, it is clear that B r ( x ) ( · ) is a lower semicontinuous func-tion. Recall that lower semicontinuity is preserved under sums, and that the supremum of an arbitrary set (not necessarily countable) of lower semicontinuous functions is also lowersemicontinuous. Therefore, from the formula N ( · , r ; X ) = sup J ′ X j ∈J ′ B r ( x j ) ( · ) , where J ′ ranges over all finite subsets of J , we deduce that the function N ( · , r ; X ) is lowersemicontinuous, and therefore measurable.Now let J R := { j ∈ J : x j ∈ B R } and note that | B r |N ( R ; X ) = X j ∈J R Z B R + r B r ( x j ) ( y ) dy . As we can interchange the sum and the integral by the monotone convergence theorem and X j ∈J R B r ( x j ) ( y ) X j ∈J B r ( x j ) ( y ) = N ( y, r ; X ) , one immediately obtains the upper bound for N ( R ; X ). Likewise, using now that | B r |N ( R ; X ) = X j ∈J R Z B R + r B r ( x j ) ( y ) dy > X j ∈J R Z B R − r B r ( x j ) ( y ) dy = X j ∈J Z B R − r B r ( x j ) ( y ) dy = Z B R − r N ( y, r ; X ) dy , we derive the lower bound. The sandwich estimate for N ( R ; X ) is then proved.Now let γ be a connected component of Γ, which is a closed set by hypothesis. Since γ ⊂ B r ( y ) if and only if y ∈ B r ( x ) for all x ∈ γ , one has that N ( y, r ; Γ) = X γ ⊂ Γ γ r ( y ) , (6.1)where the sum is over the connected components of Γ and the set γ r is defined, for eachconnected component γ of Γ, as γ r := \ x ∈ γ B r ( x ) , that is, as the set of points in R whose distance to any point of γ is less than r . Obviously,the set γ r is open, so γ r is lower semicontinuous, and contained in the ball B r ( x ), where x is any point of γ . Also notice that γ r is not the empty set provided that 2 r is largerthan the diameter of γ . Therefore, by the same argument as before, if follows from theexpression (6.1) that the function N ( · , r ; Γ) is measurable. If we now define the set Γ R consisting of the connected components of Γ that are contained in the ball B R , the same ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 29 argument as before shows that N ( R ; Γ) > X γ ⊂ Γ R | γ r | Z B R + r γ r ( y ) dy > X γ ⊂ Γ R | γ r | Z B R − r γ r ( y ) dy = X γ ⊂ Γ | γ r | Z B R − r γ r ( y ) dy > Z B R − r N ( y, r ; Γ)sup γ ⊂ Γ | γ r | dy > Z B R − r N ( y, r ; Γ) | B r | dy . In the first inequality we are summing over components γ whose diameter is smaller than2 r , and to pass to the last inequality we have used the obvious volume bound | γ r | | B r | .Note that the proof of the upper bound for N ( R ; X ) does not apply in this case, essentiallybecause we do not have lower bounds for | γ r | in terms of | B r | . (cid:3) Proof of Theorem 1.2 and Corollary 1.3.
We are ready to prove Theorem 1.2.In fact, we will establish a stronger result which permits to control the parameters of theperiodic orbits and the invariant tori. In what follows, we shall use the notation introduced inSections 4 and 5 for the number of periodic orbits N o u ( R ; [ γ ] , I ), the number of Diophantinetoroidal sets N t u ( R ; [ T ] , J , V ) (and the volume of the set of invariant tori V t u ( R ; [ T ] , J )) andthe number of horseshoes N h u ( R ). This is useful in itself, since we showed in Section 4.1that the quantity N o u ( R ; [ γ ] , I ) is finite but this does not need to be the case if one justcounts N o u ( R ; [ γ ]). Also, the choice of counting the volume of invariant tori instead of itsnumber (which one definitely expect to be infinite) provides the trivial bound V t u ( R ; [ T ] , J ) | B R | . Specifically, the result we prove is the following: Theorem 6.2.
Consider a closed curve γ and an embedded torus T of R . Then for any I = ( T , T , Λ , Λ ) , some J = ( ω , ω , τ , τ ) and some V > , where < T < T , < Λ < Λ , < ω < ω , < τ < τ , a Gaussian random Beltrami field u satisfies lim inf R →∞ N h u ( R ) | B R | > ν h , lim inf R →∞ N t u ( R ; [ T ] , J , V ) | B R | > ν t ([ T ] , J , V ) , lim inf R →∞ N o u ( R ; [ γ ] , I ) | B R | > ν o ([ γ ] , I ) with probability , with constants that are all positive. In particular, the topological entropyof u is positive almost surely, and lim inf R →∞ V t u ( R ; [ T ] , J ) | B R | > V ν t ([ T ] , J , V ) , with probability . Proof.
For the ease of notation, let us denote by Φ R ( u ) the quantities N h u ( R ), N o u ( R ; [ γ ] , I )and N t u ( R ; [ T ] , J , V ), in each case. Horseshoes are closed, and so are the set of periodicorbits isotopic to γ with parameters in I and the set of closed invariant solid tori of the kindcounted by N t u ( R ; [ T ] , J , V ). Therefore, the lower bound for sets Γ whose components areclosed proved in Lemma 6.1 ensures that, for any 0 < r < R ,Φ R ( u ) | B R | > | B R | Z B R − r Φ r ( τ y u ) | B r | dy > | B R | Z B R − r Φ mr ( τ y u ) | B r | dy , where for any large m > mr ( w ) := min { Φ r ( w ) , m } . We recall that the translation operator is defined as τ y u ( · ) = u ( · + y ).As the truncated random variable Φ mr is in L ( C k ( R , R ) , µ u ) for any m , one can considerthe limit R → ∞ and apply Proposition 3.7 to conclude thatlim inf R →∞ Φ R ( u ) | B R | > lim inf R →∞ | B R − r || B R | − Z B R − r Φ mr ( τ y u ) | B r | dy = 1 | B r | E Φ mr µ u -almost surely, for any r and m . Corollaries 4.3, 4.8 and 5.3 imply that (for any I inthe case of periodic orbits, for some J and some V > r >
0, some δ > u such that Φ r ( w ) > w ∈ C k ( R , R ) with k w − u k C ( B r ) < δ . As the randomvariable Φ r is nonnegative, and the measure µ u is supported on Beltrami fields (cf. Proposi-tion 3.8), which are divergence-free, it is then immediate that, when picking the parameters I , J and V as above, one has for k > E Φ mr > µ u (cid:0) { w ∈ C k ( R , R ) : k w − u k C k ( B r ) < δ } (cid:1) =: M ( u , δ ) . This is positive again by Proposition 3.8. So defining the constant, in each case, as ν := M ( u , δ ) | B r | > u is positive almost surely because u has a horseshoe withprobability 1, see Proposition 5.1. The estimate for the growth of the volume of Diophantineinvariant tori follows from the trivial lower bound V t u ( R ; [ T ] , J ) > V N t u ( R ; [ T ] , J , V ) . (cid:3) Remark 6.3.
A simple variation of the proof of Theorem 6.2 provides an analogous resultfor links. We recall that a link L is a finite set of pairwise disjoint closed curves in R , whichcan be knotted and linked among them. More precisely, if N l ( R ; [ L ] , I ) is the number ofunions of hyperbolic periodic orbits of u that are contained in B R , isotopic to the link L , andwhose periods and maximal Lyapunov exponents are in the intervals prescribed by I , thenlim inf R →∞ N l ( R ; [ L ] , I ) | B R | > ν l ([ L ] , I ) > . To apply the lower bound obtained in Lemma 6.1 to estimate the number of links, it is enoughto transform each link into a connected set by joining its different components by closed arcs.
ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 31
The proof then goes exactly as in Theorem 6.2 upon noticing that analogs of Proposition 4.2and Corollary 4.3 also hold for links (the proof easily carries over to this case).
Proof of Corollary 1.3.
The corollary is now an immediate consequence of the fact that thenumber of isotopy classes of closed curves and embedded tori is countable. Indeed, by Theo-rem 1.2, with probability 1, a Gaussian random Beltrami field has infinitely many horseshoes,an infinite volume of ergodic invariant tori isotopic to a given embedded torus T , and infin-itely many periodic orbits isotopic to a given closed curve γ . Since the countable intersectionof sets of probability 1 also has probability 1, the claim follows. (cid:3) Proof of Theorem 1.5.
We are now ready to prove the asymptotics for the number ofzeros of the Gaussian random Beltrami field u . Let us start by noticing that, almost surely,the zeros of u are nondegenerate. This is because µ u (cid:0)(cid:8) w ∈ C k ( R , R ) : det ∇ w ( x ) = 0 and w ( x ) = 0 for some x ∈ R (cid:9)(cid:1) = 0 , which is a consequence of the boundedness of the probability density function (cf. Remark 3.6)and that u is C ∞ almost surely, see [4, Proposition 6.5]. Hence the intersection of the zeroset X w := { x ∈ R : w ( x ) = 0 } with a ball B R is a finite set of points almost surely. The implicit function theorem thenimplies that these zeros are robust under C -small perturbations, so that with probability1, N ( R ; X v ) > N ( R ; X w ) for any vector field v that is close enough to w in the C norm.Summarizing, we have the following: Proposition 6.4.
Almost surely, the functional w
7→ N ( R ; X w ) is lower semicontinuous inthe C k compact open topology for vector fields, for any k > . Furthermore, N ( R ; X w ) < ∞ with probability . Since the variance E [ u ( x ) ⊗ u ( x )] is the identity matrix by Corollary 3.5, the Kac–Riceformula [4, Proposition 6.2] then enables us to compute the expected value of the randomvariable Φ r ( w ) := N ( r ; X w ) | B r | (6.2)as E Φ r = − Z B r E {| det ∇ w ( x ) | : w ( x ) = 0 } ρ (0) dx = (2 π ) − E {| det ∇ w ( x ) | : w ( x ) = 0 } . (6.3)Here we have used that the above conditional expectation is independent of the point x ∈ R by the translational invariance of the probability measure. We recall that the probabilitydensity function ρ ( y ) := (2 π ) − e − | y | was introduced in Remark 3.6.To compute the above conditional expectation value, one can argue as follows: Lemma 6.5.
For any x ∈ R , E {| det ∇ u ( x ) | : u ( x ) = 0 } = (2 π ) ν z , where the constant ν z is given by (1.4) . Proof.
Let us first reduce the computation of the conditional expectation to that of an ordi-nary expectation by introducing a new random variable ζ . Just like ∇ u ( x ), this new variabletakes values in the space of 3 × R by labeling thematrix entries as ζ =: ζ ζ ζ ζ ζ ζ ζ ζ ζ . (6.4)This variable is defined as ζ := ∇ u ( x ) − Bu ( x ) , (6.5)where the linear operator B (which is a 9 × ∇ u ( x ) with a vector in R )is chosen so that the covariance matrix of u ( x ) and ζ is 0: B := E ( ∇ u ( x ) ⊗ u ( x )) (cid:2) E ( u ( x ) ⊗ u ( x )) (cid:3) − = E ( ∇ u ( x ) ⊗ u ( x )) . Here we have used that the second matrix is in fact the identity by Corollary 3.5. An easycomputation shows that then E ( ζ ⊗ u ( x )) = 0 ;as u ( x ) and ζ are Gaussian vectors with zero mean, this condition ensures that they are inde-pendent random variables. Therefore, we can use the identity (6.5) to write the conditionalexpectation as E {| det ∇ u ( x ) | : u ( x ) = 0 } = E {| det[ ζ + Bu ( x )] | : u ( x ) = 0 } = E | det ζ | . Our next goal is to compute the covariance matrix of ζ in closed form, which will enableus to find the expectation of | det ζ | . By definition, E ( ζ ⊗ ζ ) = E [( ∇ u ( x ) − Bu ( x )) ⊗ ( ∇ u ( x ) − Bu ( x ))]= E [ ∇ u ( x ) ⊗ ∇ u ( x )] − E [ ∇ u ( x ) ⊗ u ( x )] E [ u ( x ) ⊗ ∇ u ( x )] . The basic observation now is that, for any Hermitian polynomials in three variables q ( ξ ) and q ′ ( ξ ), the argument that we used to establish the formula (3.3) and Corollary 3.5 shows that E [( q ( D ) u j ( x )) ( q ′ ( D ) u k ( x ))] = E [ q ( D x ) u j ( x ) q ′ ( D y ) u k ( y )] | y = x = Z S q ( ξ ) q ′ ( − ξ ) p j ( ξ ) p k ( ξ ) e iξ · ( x − y ) dσ ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) y = x = Z S q ( ξ ) q ′ ( − ξ ) p j ( ξ ) p k ( ξ ) dσ ( ξ ) . Here we have used that q ′ ( D ) u k is real-valued because q ′ is Hermitian. As all the matrixintegrals in the calculation of E ( ζ ⊗ ζ ) are of this form with q ( ξ ) = iξ or 1, the computationagain boils down to evaluating integrals of the form R S ξ α dσ ( ξ ), which can be computed usingthe formula (3.4). ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 33
Tedious but straightforward computations then yield the following explicit formula for thecovariance matrix of ζ :Σ := E ( ζ ⊗ ζ ) = − − − −
00 0 − − Note that this matrix is not invertible: it has rank 5, and an orthogonal basis for the (4-dimensional) kernel is { e + e + e , e − e , e − e , e − e } , where { e j } j =1 denotes the canonical basis of R . As we are dealing with Gaussian vectors,this is equivalent to the assertion that ζ + ζ + ζ = 0 , ζ = ζ , ζ = ζ , ζ = ζ (6.6)almost surely (which amounts to saying that ζ is a traceless symmetric matrix). Noticethat these equations define a 5-dimensional subspace orthogonal to the kernel of Σ. Theremaining random variables ζ ′ := ( ζ , ζ , ζ , ζ , ζ ) are independent Gaussians with zero meanand covariance matrixΣ ′ := E ( ζ ′ ⊗ ζ ′ ) = − −
00 0 0 0
By construction, Σ ′ is an invertible matrix, so we can immediately write down a formula forthe expectation value of | det ζ | : E | det ζ | = (2 π ) − (det Σ ′ ) − Z R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) det ζ ζ ζ ζ ζ ζ ζ ζ − ζ − ζ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e − ζ ′ · Σ ′− ζ ′ dζ ′ = (2 π ) − (det Σ ′ ) − Z R | Q ( ζ ′ ) | e − ζ ′ · Σ ′− ζ ′ dζ ′ , with the cubic polynomial Q being defined as in (1.5). Since ζ ′ · Σ ′− ζ ′ = e Q ( ζ ′ ), where thequadratic polynomial e Q was defined in (1.6), anddet Σ ′ = 5 · · , we therefore have E | det ζ | = (2 π ) ν z . The result then follows. (cid:3)
Remark 6.6.
If one keeps track of the connection between ζ and ∇ u ( x ), it is not hard tosee that the first condition ζ + ζ + ζ = 0 in (6.6) is equivalent to div u ( x ) = 0, while theremaining three just mean that curl u ( x ) = u ( x ), at the points x ∈ R where u ( x ) = 0. In particular, this shows that Φ R ∈ L ( C k ( R , R ) , µ u ). For the ease of notation, let usdefine the ergodic mean operator A R Φ( w ) := 1 | B R | Z B R Φ( τ y w ) dy . Since N ( R, X w ) is finite almost surely, cf. Proposition 6.4, the sandwich estimate proved inLemma 6.1 implies that, almost surely,1 | B R | Z B R − r Φ r ( τ y w ) dy Φ R ( w ) | B R | Z B R + r Φ r ( τ y w ) dy for any 0 < r < R . Therefore, and using that | B R ± r | / | B R | = (1 ± r/R ) , one has | Φ R − A R Φ r | (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) rR (cid:19) A R + r Φ r − A R Φ r (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) − rR (cid:19) A R − r Φ r − A R Φ r (cid:12)(cid:12)(cid:12)(cid:12) . For fixed r , Equation (6.3) and Proposition 3.7 ensure that A R Φ r L −−→ a . s . E Φ r = ν z (6.7)as R → ∞ ; also, note that the limit (which is independent of r ) has been computed inLemma 6.5 above.Therefore, if we let R → ∞ while r is held fixed, the RHS of the estimate before Equa-tion (6.7) tends to 0 µ u -almost surely and in L ( µ u ), so thatΦ R − A R Φ r L −−→ a . s . R → ∞ . As A R Φ r L −−→ a . s . ν z by (6.7), Theorem 1.5 is proven.7. The Gaussian ensemble of Beltrami fields on the torus
Gaussian random Beltrami fields on the torus.
As introduced in Section 1.3, aBeltrami field on the flat 3-torus T := ( R / π Z ) (or, equivalently, on the cube of R of sidelength 2 π with periodic boundary conditions) is a vector field on T satisfying the equationcurl v = λv for some real number λ = 0. To put it differently, Beltrami fields on the torus are theeigenfields of the curl operator. It is easy to see that such an eigenfield is divergence-free andhas zero mean, that is, R T v dx = 0.Since ∆ v + λ v = 0, it is well-known (see e.g. [10]) that the spectrum of the curl operatoron the 3-torus consists of the numbers of the form λ = ±| k | for some vector with integercoefficients k ∈ Z . For concreteness, we will henceforth assume that λ >
0; the case ofnegative frequencies is completely analogous. Since k has integer coefficients, one can labelthe positive eigenvalues of curl by a positive integer L such that λ L = L / . Let us define Z L := { k ∈ Z : | k | = L } and note that the set Z L is invariant under reflections (i.e., − k ∈ Z L if k ∈ Z L ).The Beltrami fields corresponding to the eigenvalue λ L must be of the form v = X k ∈Z L V k e ik · x , (7.1) ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 35 for some V k ∈ C . Conversely, this expression defines a Beltrami field with frequency λ L ifand only if V k = V − k (which ensures that v is real valued) and ikL / × V k = V k . Since | k | = L / , we infer from the proof of Proposition 2.1 that the vector V k must be of theform V k = α k p ( k/L / ) (7.2)unless k = ( ± L / , , α k ∈ C is an arbitrary complex number and the Hermitianvector field p ( ξ ) was defined in (2.4).The multiplicity of the eigenvalue λ L is given by the cardinality d L := Z L . By Legendre’sthree-square theorem, Z L is nonempty (and therefore λ L is an eigenvalue of the curl operator)if and only if L is not of the form 4 a (8 b + 7) for nonnegative integers a and b .Based on the formulas (7.1)-(7.2), we are now ready to define a Gaussian random Beltramifield on the torus with frequency λ L as u L ( x ) := (cid:18) πd L (cid:19) / X k ∈Z L a Lk p ( k/L / ) e ik · x , (7.3)where the real and imaginary parts of the complex-valued random variable a Lk are standardGaussian variables. We also assume that these random variables are independent except forthe constraint a Lk = a L − k . The inessential normalization factor (2 π/d L ) / has been introducedfor later convenience.Note that u L ( x ) is a smooth R -valued function of the variable x , so it induces a Gaussianprobability measure µ L on the space of C k -smooth vector fields on the torus, C k ( T , R ). Asbefore, we will always assume that k > u L,z ( x ) := u L (cid:18) z + xL / (cid:19) for any fixed point z ∈ T .7.2. Estimates for the rescaled covariance matrix.
In what follows, we will restrict ourattention to the positive integers L , which we will henceforth call admissible , that are notcongruent with 0, 4 or 7 modulo 8. When L is congruent with 0 or 7 modulo 8, Legendre’sthree-square theorem immediately implies that Z L is empty. The reason to rule out numberscongruent with 4 modulo 8 is more subtle: a deep theorem of Duke [9], which addresses aquestion raised by Linnik, ensures that the set Z L /L / becomes uniformly distributed onthe unit sphere as L → ∞ through integers that are congruent to 1, 2, 3, 5 or 6 modulo 8.This ensures that 4 πd L X k ∈Z L φ ( k/L / ) → Z S φ ( ξ ) dσ ( ξ ) (7.4)as L → ∞ through admissible values, for any continuous function φ on S . A particular caseis when L goes to infinity through squares of odd values, that is, when L = (2 m + 1) and m → ∞ .The covariance kernel of the Gaussian random variable u L is the matrix-valued function κ L ( x, y ) := E L [ u L ( x ) ⊗ u L ( y )] . Following Nazarov and Sodin [30], we will be most interested in the covariance kernel of therescaled field u L,z at a point z ∈ T , which is given by κ L,z ( x, y ) = E L (cid:20) u L (cid:18) z + xL / (cid:19) ⊗ u L (cid:18) z + yL / (cid:19)(cid:21) . The following proposition ensures that, for large admissible frequencies L , the rescaled co-variance kernel, and suitable generalizations thereof, tend to those of a Gaussian randomBeltrami field on R , κ ( x, y ), defined in (3.2): Proposition 7.1.
For any z ∈ T , the rescaled covariance kernel κ L,z ( x, y ) has the followingproperties: (i) It is invariant under translations and independent of z . That is, there exists somefunction κ L such that κ L,z ( x, y ) = κ L ( x − y ) . (ii) Given any compact set K ⊂ R , the covariance kernel satisfies κ L,z ( x, y ) → κ ( x, y ) in C s ( K × K ) as L → ∞ through admissible values.Proof. Let α , β be any multiindices, and recall the operator D = − i ∇ introduced in Section 3.By definition, and using the fact that u L is real, D αx D βy κ L,z ( x, y ) = E L (cid:20) D αx u L (cid:18) z + xL / (cid:19) ⊗ D βy u L (cid:18) z + yL / (cid:19)(cid:21) = E L (cid:20) D αx u L (cid:18) z + xL / (cid:19) ⊗ D βy u L (cid:18) z + yL / (cid:19)(cid:21) = 2 πd L X k ∈Z L X k ′ ∈Z L E L ( a Lk a Lk ′ ) p (cid:18) kL / (cid:19) ⊗ p (cid:18) k ′ L / (cid:19) (cid:18) kL / (cid:19) α (cid:18) − k ′ L / (cid:19) β e ik · ( z + xL / ) − ik ′ · ( z + yL / ) . The independence properties of the Gaussian variables a Lk (which have zero mean) imply that E L ( a Lk a Lk ′ ) = 0 if k ′
6∈ { k, − k } . When k ′ = k one has E L [ | a Lk | ] = E L [(Re a Lk ) ] + E L [(Im a Lk ) ] = 2 , and when k ′ = − k , E L [( a Lk ) ] = E L [(Re a Lk ) ] − E L [(Im a Lk ) ] + 2 i E L [(Re a Lk )(Im a Lk )] = 0 . Therefore, E L ( a Lk a Lk ′ ) = 2 δ kk ′ and we obtain D αx D βy κ L,z ( x, y ) = 4 πd L X k ∈Z L p (cid:18) kL / (cid:19) ⊗ p (cid:18) kL / (cid:19) (cid:18) kL / (cid:19) α (cid:18) − kL / (cid:19) β e ik · ( x − y ) /L / . In particular, this formula shows that κ L,z ( x, y ) is independent of z and translation-invariant.Using now the fact that Z L becomes uniformly distributed on S as L → ∞ throughadmissible values, we obtain via Equation (7.4) that D αx D βy κ L,z ( x, y ) → Z S ξ α ( − ξ ) β p ( ξ ) ⊗ p ( ξ ) e iξ · ( x − y ) dσ ( ξ )= D αx D βy Z S p ( ξ ) ⊗ p ( ξ ) e iξ · ( x − y ) dσ ( ξ ) . By Proposition 3.4, the RHS equals D αx D βy κ ( x, y ), so the result follows. (cid:3) ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 37
A convergence result for probability measures.
We shall next present a resultshowing that the probability measure defined by the rescaled field u L,z converges, as L → ∞ ,to that defined by the Gaussian random Beltrami field on R , u , on compact sets of R : Lemma 7.2.
Fix some
R > and denote by µ L,zR and µ u,R , respectively, the probabilitymeasures on C k ( B R , R ) defined by the Gaussian random fields u L,z and u . Then the mea-sures µ L,zR converge weakly to µ u,R as L → ∞ through the admissible integers.Proof. Let us start by noting that all the finite dimensional distributions of the fields u L,z converge to those of u as L → ∞ . Specifically, consider any finite number of points x , . . . , x n ∈ R , any indices j , . . . , j n ∈ { , , } , and any multiindices with | α j | k . Thenit is not hard to see that the Gaussian vectors of zero expectation( ∂ α u L,zj ( x ) , . . . , ∂ α n u L,zj n ( x n )) ∈ R n converge in distribution to the Gaussian vector( ∂ α u j ( x ) , . . . , ∂ α n u j n ( x n )) (7.5)as L → ∞ . This follows from the fact that their probability density functions are completelydetermined by the n × n variance matrixΣ L := (cid:16) ∂ α l x ∂ α m y κ L,zj l j m ( x, y ) (cid:12)(cid:12) ( x,y )=( x l ,x m ) (cid:17) l,m n , which converges to Σ := ( ∂ α l x ∂ α m y κ j l j m ( x, y ) | ( x,y )=( x l ,x m ) ) as L → ∞ by Proposition 7.1. Thelatter, of course, is the covariance matrix of the Gaussian vector (7.5).It is well known that this convergence of arbitrary Gaussian vectors is not enough toconclude that µ L,zR converges weakly to µ u,R . However, notice that, for any integer s >
0, themean of the H s -norm of u L,z is uniformly bounded: E L,z k w k H s ( B R ) = X | α | s E Z B R | D α u L,z ( x ) | dx = X | α | s Z B R tr (cid:16) D αx D αy κ L,z ( x, y ) (cid:12)(cid:12) y = x (cid:17) dx −−−−→ L →∞ X | α | s Z B R tr (cid:16) D αx D αy κ ( x, y ) (cid:12)(cid:12) y = x (cid:17) dx < M s,R . To pass to the last line, we have used Proposition 7.1 once more. As the constant M s,R isindependent of L , Sobolev’s inequality ensures thatsup L E L,z k w k C k +1 ( B R ) C sup L E L,z k w k H k +3 ( B R ) < M for some constant M that only depends on R . For any ε >
0, this implies that for alladmissible L large enough µ L,zR (cid:0)(cid:8) w ∈ C k ( B R , R ) : k w k C k +1 ( B R ) > M/ε (cid:9)(cid:1) < ε . As the set { w ∈ C k ( B R , R ) : k w k C k +1 ( B R ) M/ε } is compact by the Arzel`a–Ascoli theorem,we conclude that the sequence of probability measures µ L,zR is tight. Therefore, a straight-forward extension to jet spaces of the classical results about the convergence of probabilitymeasures on the space of continuous functions [7, Theorem 7.1], carried out in [34], per-mits to conclude that µ L,zR indeed converges weakly to µ u,R as L → ∞ . The lemma is thenproven. (cid:3) Proof of Theorem 1.6.
We are now ready to prove our asymptotic estimates forhigh-frequency Beltrami fields on the torus. The basic idea is that, by the definition of therescaling, µ L (cid:0)(cid:8) w ∈ C k ( T , R ) : N h w > m (cid:9)(cid:1) > µ L,zR (cid:0)(cid:8) w ∈ C k ( B R , R ) : N h w ( r ) > m (cid:9)(cid:1) provided that r < R < L / : this just means that the number of horseshoes that u L has inthe whole torus is certainly not less than those that are contained in a ball centered at anygiven point z ∈ T of radius r/L / <
1. The same is clearly true as well when one countsinvariant solid tori, periodic orbits or zeros instead.For the ease of notation, let us denote by Φ r ( w ) the quantity N h w ( r ), N t w ( r ; [ T ] , J , V ), N o w ( r ; [ γ ] , I ) or N z w ( r ) (that is, the number of nondegenerate zeros of w in B r ), in eachcase. See Sections 4 and 5 for precise definitions. We recall that N z w ( r ) = N ( r ; X w ) withprobability 1, cf. Section 6.3. Theorems 6.2 (for periodic orbits, invariant tori and horseshoes)and 1.5 (for zeros) ensure that, given any m >
0, any δ >
0, any closed curve γ and anyembedded torus T , one can find some parameters I , J , V and r > µ u (cid:0)(cid:8) w ∈ C k ( R , R ) : Φ r ( w ) > m (cid:9)(cid:1) > − δ . Of course, here we are simply using that the volume | B r | , which appears in the statementsof Theorems 6.2 and 1.5 but not here, can be made arbitrarily large by taking a large r .Let us now fix some R > r and some point z ∈ T . We showed in Propositions 4.1, 4.6, 5.1and 6.4 that the functionals that we are now denoting by Φ r are lower semicontinuous on thespace C k ( R , R ) of divergence-free fields for k >
4. This implies that the setΩ r,R,m := { w ∈ C k ( B R , R ) : Φ r ( w ) > m } is open in C k ( B R , R ). Lemma 7.2 ensures that the measure µ L,zR converges weakly to µ u,R as L → ∞ through the admissible integers. As the set Ω r,R,m is open, this is well known toimply (see e.g. [7, Theorem 2.1.iv]) thatlim inf L →∞ µ L,zR (Ω r,R,m ) > µ u,R (Ω r,R,m )= µ u (cid:0)(cid:8) w ∈ C k ( R , R ) : Φ r ( w ) > m (cid:9)(cid:1) > − δ . We observe that δ > r is large enough (and r/L /
1. Since the previous estimate ensures that N t w ( r ; [ T ] , J , V ) > m with probability 1 as L → ∞ , we infer that the probability of havingan infinite number of (Diophantine) invariant tori isotopic to T also tends to 1 as L → ∞ through the admissible integers. However this does not provide any information about theexpected volume of the invariant tori.The result about the topological entropy follows from the following observation. If wedenote by φ Lt the time- t flow of the Beltrami field u L ( z + · ), and by φ t the flow of the rescaled ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 39 field u L,z , it is evident that φ Lt = 1 L / φ L / t . Then, the topological entropy h top ( u L ), which is defined as the entropy of its time-1 flow,satisfies h top ( u L ) = h top ( φ L ) = h top (cid:16) L / φ L / (cid:17) = h top ( φ L / ) = L / h top ( φ ) (7.6)= L / h top ( u L,z ) . (7.7)In the third equality we have used that the topological entropy does not depend on the spacescale (or equivalently, on the metric), and in the fourth equality we have used Abramov’swell-known formula (see e.g. [19]). Since the rescaled field has a horseshoe in a ball of radius r with probability 1 as L → ∞ , and a horseshoe has positive topological entropy, say largerthan some constant ν h ∗ (see Proposition 5.1), Equation (7.6) implies that the topologicalentropy of u L is at least ν h ∗ L / .Finally, we prove the statement about the expected values. As above, we use the functionalΦ r ( w ) to denote the number of different objects (horseshoes, solid tori or periodic orbits).The case of zeros will be considered later. Note that, since Φ r is lower semicontinuous, and µ L,z converges weakly to µ u as L → ∞ by Lemma 7.2, it is standard that [7, Exercise 2.6]lim inf L →∞ E L,z Φ r | B r | > E Φ r | B r | > η > , where we have picked some fixed, large enough r . Here we have used the asymptotics in R ,given by Theorem 6.2, to infer that the last expectation is positive if r is large. Notice that theconstant η depends on [ γ ] , [ T ] , I or J depending on the functional the we are considering,but we shall not write this dependence explicitly. Furthermore, as the distribution of themeasure µ L,zR is in fact independent of z by Proposition 7.1, this ensures that there is some L independent of z such that E L,z Φ r | B r | > η L > L and all z ∈ T .Now, given any admissible L > L , it is standard that we can cover the torus T by balls { B r L ( z a ) : 1 a A L } of radius r L := 2 r/L / centered at z a ∈ T such that the smallerballs B r L / ( z a ) are pairwise disjoint. This implies that A L > cL for some dimensionalconstant c . The expected value of, say, the number of horseshoes of u L in T can then becontrolled as follows, for any admissible L > L : E L N h L / > A L X a =1 | B r | L / E L,z a Φ r | B r | > c | B r | η > ν ∗ for some positive constant ν ∗ independent of L . An analogous estimate holds for the expectedvalue E L N o ([ γ ]). To estimate the volume of ergodic invariant tori isotopic to T we can proceed as follows.For any admissible L > L we have: E L V t ([ T ]) > A L X a =1 | B r L / | E L,z a V t ( r ; [ T ] , J ) | B r | > A L X a =1 | B r L / | V E L,z a Φ r | B r | > V η A L X a =1 | B r L / | > ν t ∗ ([ T ])for some positive constant ν t ∗ ([ T ]) independent of L . Here we have used that the balls B r L / ( z a ) are pairwise disjoint and the sum of their volumes is, by construction, largerthan | T | / Lemma 7.3. E L ( L − N z u L ) → (2 π ) ν z as L → ∞ through admissible values.Proof. Let us use the notation Q R := ( − Rπ, Rπ ) × ( − Rπ, Rπ ) × ( − Rπ, Rπ )for the open cube of side 2 πR in R and call N z , ∗ u L the number of zeros of u L (or rather of itsperiodic lift to R ) that are contained in Q . By Bulinskaya’s lemma [4, Proposition 6.11],with probability 1 the zero set of u L is nondegenerate (and hence a finite set of points) andthe lift of u L does not have any zeros on the boundary ∂Q . Therefore, for any positiveinteger R , N z u L = N z , ∗ u L almost surely. In particular, both quantities have the same expectation.Let us now take some small positive real r and denote by N z u L ( y, r ) the number of zeros of u L (or rather of its lift to R ) that are contained in the ball B r ( y ). The argument we used toprove the estimate for N ( R ; X ) in Lemma 6.1 (starting now from the number of zeros in Q instead of in B R ) shows that Z Q − r N z u L ( z, r ) | B r | dz N z , ∗ u L Z Q r N z u L ( z, r ) | B r | dz . Note now that Z Q ± r N z u L ( z, r ) | B r | dz = L Z Q ± r N z u L,z ( rL / ) | B rL / | dz . The expected value of this quantity is E L Z Q ± r N z u L,z ( rL / ) | B rL / | dz = Z Q ± r E L,z N z u L,z ( rL / ) | B rL / | dz = | Q ± r | E L,z N z u L,z ( rL / ) | B rL / | . To pass to the second line we have used that the expected value inside the integral is in-dependent of the point z by Proposition 7.1; in particular, this value is independent of thepoint z one considers. ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 41
We can now argue just as in the case of R , discussed in detail in Subsection 6.3, so wewill just sketch the arguments and refer to that subsection for the notation. The Kac–Riceformula ensures E L,z N z u L,z ( rL / ) | B rL / | = (2 π ) − E L,z (cid:0)(cid:8) | det ∇ u L,z (0) | : u L,z (0) = 0 (cid:9)(cid:1) , and this conditional expectation can be transformed into an unconditional one just as in theproof of Lemma 6.5: E L,z N z u L,z ( rL / ) | B rL / | = (2 π ) − / E L,z ( | det ζ L,z | )= (2 π ) − / (2 π ) / (det Σ ′ L,z ) / Z R Q L,z ( ζ ′ ) e − ζ ′ · (Σ ′ L,z ) − ζ ′ dζ ′ =: ν z ,L,z . The fact that the covariance matrix of u L,z converges to that of u as L → ∞ by Proposition 7.1implies that lim L →∞ ν z ,L,z = ν z . Hence, writing the aforementioned sandwich estimate as | Q − r | ν z ,L,z E L N z u L L / | Q r | ν z ,L,z and letting L → ∞ and then r →
0, we infer thatlim L →∞ E L N z u L L / = | Q | ν z = (2 π ) ν z . The lemma follows. (cid:3)
Theorem 1.6 is then proven.
Acknowledgements
The authors are deeply indebted to Alejandro Luque for the numerical computation ofthe Melnikov constants in Section 5. A.E. is supported by the ERC Starting Grant 633152.D.P.-S. is supported by the grants MTM-2016-76702-P (MINECO/FEDER) and EuropaExcelencia EUR2019-103821 (MCIU). A.R. is supported by the grant MTM-2016-76702-P(MINECO/FEDER). This work is supported in part by the ICMAT–Severo Ochoa grantSEV-2015-0554 and the CSIC grant 20205CEX001.
Appendix A. Fourier-theoretic characterization of Beltrami fields
For the benefit of the reader, in this appendix we describe what polynomially bounded Bel-trami fields look like in Fourier space. As Beltrami fields are a particular class of vector-valuedmonochromatic waves, it is convenient to start the discussion by considering polynomiallybounded solutions to the Helmholtz equation∆ F + F = 0 . As before, we consider the case of monochromatic waves on R , but the analysis appliesessentially verbatim to any other dimension. The Fourier transform of this equation showsthat (1 − | ξ | ) b F ( ξ ) = 0 , so the support of b F must be contained in the unit sphere, S . In spherical coordinates ρ := | ξ | ∈ R + and ω := ξ/ | ξ | ∈ S , it is standard that this is equivalent to saying that b F is a finitesum of the form b F = N X n =1 F n ( ω ) δ ( n ) ( ρ − . Here δ ( n ) is the n th derivative of the Dirac measure and F n is a distribution on the sphere, so F n ∈ H s n ( S ) for some s n ∈ R (because any compactly supported distribution is in a Sobolevspace, possibly of negative order). Note that F is real valued if and only if the functions F n are Hermitian. Of course, there are also monochromatic waves that are not polynomiallybounded, such as F := e x cos( √ x ).A classical result due to Herglotz [26, Theorem 7.1.28] ensures that if F is a monochromaticwave with the sharp fall off at infinity, i.e., such thatlim sup R →∞ R Z B R F dx < ∞ , then there is a Hermitian vector-valued function f ∈ L ( S ) such that b F = f δ ( ρ − C k F k L ( S ) , C k F k L ( S ) ] for some con-stants C , C . This bound means that, on an average sense, | F ( x ) | decays as C/ | x | . The primeexample of this behavior is given by f = 1, which corresponds to F ( x ) = c | x | − / J / ( | x | ).The expression (1.3) corresponds to the case N = 0 above, since the function F with b F = f ( ω ) δ ( ρ −
1) is precisely F ( x ) = Z S e ix · ω f ( ω ) dσ ( ω ) . Also, if f ∈ H − k ( S ) with k > L ( S ), the function F is boundedas [15, Appendix A] sup R> R Z B R F ( x ) | x | k dx C k f k H − k ( S ) . (A.1)Hence in this case, F is bounded, on an average sense, by C | x | k − . Therefore, if f ∈ H − ( S ), F is uniformly bounded in average sense.If f is a Gaussian random field, as considered in the Nazarov–Sodin theory (see Equa-tion (1.3a)), we showed in Proposition 3.2 that f is almost surely in H − − δ ( S ) for all δ > L ( S ). This behavior morally corresponds to functions that are bounded on a av-erage sense but do not decay at infinity, as illustrated by the function F := cos x generatedby f := [ δ ξ + ( ξ ) + δ ξ − ( ξ )]. This is the kind of behavior one needs to describe the expectedlocal behavior of a high energy eigenfunction on a compact manifold as one zooms in at agiven point.The monochromatic wave defined as b F n := f ( ω ) δ ( n ) ( ρ −
1) reads, in physical space, as F n ( x ) = Z S Z ∞ e iρx · ω f ( ω ) ρ δ ( n ) ( ρ − dρ dσ ( ω ) = ( − n Z S f ( ω ) ∂ nρ | ρ =1 ( ρ e iρx · ω ) dσ ( ω ) . Note that the n th derivative term involves an n th power of x . Therefore, using the bound (A.1),one easily finds that F n is bounded on average as C | x | n + k − if f ∈ H − k ( S ); explicit exampleswith this growth can be easily constructed by taking f to be either a constant for k = 0 orthe ( k − th derivative of the Dirac measure for k >
1. Consequently, picking f as in (1.3a),the bound (A.1) morally leads to thinking of F n as a function that grows as | x | n at infinity,which cannot be the localized behavior of an eigenfunction. This is the rationale for defining ELTRAMI FIELDS EXHIBIT KNOTS AND CHAOS 43 a random monochromatic wave as in (1.3a)-(1.3b). In this direction, let us recall that therelation between random monochromatic waves and zoomed-in high energy eigenfunctions ona various compact manifolds is an influential long-standing conjecture of Berry [6]. A preciseform of this relation has been recently established in the case of the round sphere and ofthe flat torus [29, 30, 31], which heuristically shows that (1.3a)-(1.3b) is indeed the properdefinition of random monochromatic waves for this purpose.The reasoning leading to the definition of a random Beltrami field as (1.3) is completelyanalogous, and the fact that one can relate Gaussian random Beltrami fields on R to high-frequency Beltrami fields on the torus just as in the case of the Nazarov–Sodin theory heuris-tically ensures that this is indeed the appropriate definition. For completeness, let us recordthat, just as in the case of monochromatic random waves, the Fourier transform of a polyno-mially bounded Beltrami field u is a finite sum of the form b u = N X n =1 f n ( ω ) δ ( n ) ( ρ − , where now f n is a Hermitian C -valued distribution on S . For u to be a Beltrami field, thereis an additional constraint on f n coming from the fact that not every distribution supportedon S satisfies the equation iξ × b u ( ξ ) = b u ( ξ ). A straightforward computation shows that thisconstraint amounts to imposing that N X n = j (cid:18) nj (cid:19) α n − j, f n ( ω ) = iω × N X n = j (cid:18) nj (cid:19) α n − j, f n ( ω )on S for all 0 j N . Here α k,l := Q k − m =0 ( l − m ) with the convention that α ,l := 1. To seethis, it suffices to note that the action of b u and iξ × b u on a vector field w ∈ C ∞ c ( R , R ) is h b u, w i = N X n =0 ( − n Z S f n ( ω ) · ∂ nρ | ρ =1 (cid:2) ρ w ( ρω ) (cid:3) dσ ( ω ) , h iξ × b u, w i = N X n =0 ( − n Z S iω × f n ( ω ) · ∂ nρ | ρ =1 (cid:2) ρ w ( ρω ) (cid:3) dσ ( ω ) , expand the n th derivative using the binomial formula and note that α k,l is the k th derivativeof ρ l at ρ = 1. References [1] V.I. Arnold, Sur la topologie des ´ecoulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris261 (1965) 17–20.[2] V.I. Arnold, Sur la g´eom´etrie diff´erentielle des groupes de Lie de dimension infinie et ses applications `al’hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966) 319–361.[3] V.I. Arnold, B. Khesin,
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Instituto de Ciencias Matem´aticas, Consejo Superior de Investigaciones Cient´ıficas, 28049Madrid, Spain
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