OON THE SPECTRUM OF 1D QUANTUM ISINGQUASICRYSTAL
WILLIAM N. YESSEN
Abstract.
We consider one dimensional quantum Ising spin-1 / Introduction
Since the discovery of quasicrystals [49,50,64,69], quasi-periodic models in math-ematical physics have formed an active area of research. In all of these mod-els, quasi-periodicity is introduced via so-called quasi-periodic sequences, which,roughly speaking, are somewhat intermediate between periodic and random (fora textbook exposition, see [25, 59]), and are meant to model microscopic quasi-periodic structure of quasicrystals (not all such sequences necessarily correspondto a microscopic organization of an actual physical material, but the Fibonaccisubstitution sequence, which we consider here, does correspond to an actual qua-sicrystal made up of two microscopic constituents whose arrangement resemblesthe Fibonacci substitution sequence). One of the main tools in the investigationof such models has been renormalization analysis, leading to renormalization mapswhose properties (or, better to say, their action on appropriate spaces, typically R n ) yield strong implications for the underlying models, which are usually verydifficult or impossible to obtain by other means. These renormalization maps havebeen called trace maps (roughly) since they were originally introduced in [45,48,55](see also [3, 37, 44, 47, 62, 63, 70, 75] and references therein). The terminology ismeant to emphasize their intimate connection to the transfer matrix formalism,perhaps better known to statistical physicists, which is a renormalization proce-dure that allows one to study some relevant properties of statistical-mechanicalmodels via traces of appropriate transfer operators . (One has to note, however,that in the past few decades these techniques have been greatly generalized by Date : November 9, 2018.2010
Mathematics Subject Classification.
Primary: 82B20, 82B44, 82D30. Secondary: 82D40,82B10, 82B26, 82B27.
Key words and phrases. lattice systems, disordered systems, quasi-periodicity, quasicrystals,disordered materials, quantum Ising model, Fibonacci Hamiltonian, trace map.The author was supported by the NSF grants DMS-0901627, PI: A. Gorodetski and IIS-1018433, PI: M. Welling and Co-PI: A. Gorodetski. a r X i v : . [ m a t h - ph ] A p r W. N. YESSEN mathematicians, leading to exciting results in a few fields; among the wider knownones would be spectral theory [in particular of ergodic Schr¨odinger operators] anddynamical systems [in the spirit of works by R. Bowen, D. Ruelle and Ya. Sinaion thermodynamic formalism]). The method of trace maps has led, for example,to fundamental results in spectral theory of discrete Schr¨odinger operators andIsing models on one-dimensional quasi-periodic lattices (for Schr¨odinger operators:[5, 12, 14, 15, 17, 18, 20, 23, 48, 60, 71], for Ising models: [6, 7, 13, 24, 33, 74, 76, 84], andreferences therein).Quasi-periodicity and the associated trace map formalism is still an area ofactive investigation, mostly in connection with their applications in physics. In thispaper we use these techniques to investigate the energy spectrum of one-dimensionalquantum Ising spin chains with two-valued nearest neighbor couplings arrangedin a quasi-periodic sequence, with uniform, transverse magnetic field. We shallconcentrate on the quasi-periodic sequence generated by the Fibonacci substitutionon two symbols, which (probably due to the original choice of models in early 1980’s)is the most widely studied case (and a representative example of many observedphenomena exhibited by quasi-periodic models in general).One-dimensional quasi-periodic quantum Ising spin chains have been investi-gated (analytically and numerically) over the past two decades [3,6,7,13,24,33,39–42,77,84]. Numerical and some analytic results suggest Cantor structure of the en-ergy spectrum, with nonuniform local scaling (i.e. a multifractal). The multifractalstructure of the energy spectrum has not been shown rigorously.Our aim here is to prove the multifractality of the energy spectrum and inves-tigate its fractal dimensions. We achieve this by carrying out a renormalizationprocedure, using in a crucial way quasi-periodicity of (i.e. the repetitive, albeitnot periodic, nature of) the sequence of interaction couplings (which are chosen ac-cording to the aforementioned Fibonacci substitution). The renormalization mapturns out to be a second degree polynomial acting on R – the so-called trace map .We then investigate the discrete time dynamics of this polynomial map, relatingthe energy spectrum to invariant sets for the renormalization map. Techniquesfrom hyperbolic and partially hyperbolic dynamics are then employed to inves-tigate topological and fractal-dimensional properties of these invariant sets, con-cluding with our main result about the energy spectrum which, roughly speaking,states: The spectrum is a zero measure Cantor set with non-constant local scaling(i.e. a multifractal); the spectrum and its fractal dimensions depend continuouslyon the parameters of the model . We should note that these observations are notentirely in line with what has been observed in spectral analysis of quasi-periodicSchr¨odinger operators, since in the latter, the spectrum, while a Cantor set, is nota multifractal (i.e. local scaling is uniform throughout the spectrum). We commenton this in more detail in Section 5.1.Certainly the technique described above is not new and has been employed nu-merous times (see the references above), most notably in the context of quasi-periodic Schr¨odinger operators in one dimension; however, to the best of the au-thor’s knowledge, this is the first rigorous treatment of quantum one-dimensionaldiscrete quasi-periodic Ising models that provides rigorous results regarding themultifractal structure of the spectrum. Even though, as is mentioned later in thecourse of this paper, the Ising model that we are concerned with here can be mappedcanonically to a Jacobi operator (which resembles in many ways the Schr¨odinger
PECTRUM OF QUANTUM ISING QUASICRYSTAL 3 operator), the method of trace maps, while still applicable, presents certain tech-nical difficulties that are not present in the context of Schr¨odinger operators. Wediscuss in more detail these difficulties in Section 5.1; let us only mention here thatthe techniques that we developed in this paper seem to be applicable in a widerange of models, and can be substantially generalized and presented in a model-independent fashion. The present paper, however, is already rather involved andtechnical (which is to be expected, as one has to perform a number of transforma-tions from the original spectral-theoretic problem to a problem of geometry anddynamical systems); for this reason it had been decided to postpone generaliza-tions (at the time when this paper was written). First steps towards developinga general toolbox were later taken in [21, Section 2] jointly with D. Damanik andP. Munger. In that same paper, [21], we investigated spectral properties of a classof five-diagonal unitary operators, commonly called the
CMV matrices , that playa central role in the theory of orthogonal polynomials on the unit circle (in fact,CMV matrices are to orthogonal polynomials on the unit circle as the tridiagonalJacobi operators are to orthogonal polynomials on the real line [66, 67]). Not sur-prisingly, the techniques that we develop here (and generalize in [21]), among othertechniques, were successfully applied to spectral theory of quasi-periodic CMV ma-trices in [21]. In the same fashion, these techniques have been successfully appliedto quasi-periodic Jacobi matrices in [82]. Applications of our results from [21] willbe appearing in our forthcoming paper [22]; among the applications are the quan-tum walks on one-dimensional lattices with quasi-periodic (Fibonacci in particular)coin flips, and a canonical transformation between classical one-dimensional Isingmodels with external magnetic field and CMV matrices.Not to get sidetracked too far, let us conclude this introduction by noting thatwe study and apply the trace map as a real analytic map ; study of the complexifiedversion (though within a different context) was carried out by S. Cantat in [11].We have applied the trace map as a holomorphic map on C in [80] as a renor-malization map for the classical one-dimensional Ising model with quasi-periodicnearest neighbor interaction and magnetic field, and were able to relate the analyt-icity of the free energy function to the analyticity of the escape rate of orbits underthe action of the trace map, proving absence of phase transitions. Also in [80] weapplied the techniques from the present paper to obtain precise description of theLee-Yang zeros of the classical model in the thermodynamic limit. (The results in[80] were obtained some months after the present paper was written). At this pointwe would like to mention the work of P. Bleher et. al. [8–10] on Ising models oncertain two-dimensional lattices (though not quasi-periodic and not involving tracemaps), where the action of the renormalization map was treated as a holomorphicdynamical system.Let us also briefly mention that, apart from their applications, trace maps presentan interest to the dynamical systems community as a family of polynomial mapsexhibiting quite rich dynamics ([11,38,61–63] and references therein, and our forth-coming work [28]). In addition to the references from above, for a broad overviewof the recent developments and open problems, the interested reader may consultthe surveys [16] (with emphasis on the Schr¨odinger operator) and the forthcoming[81] (with emphasis on the dynamics of trace maps and applications to a class ofmodels, including quantum and classical Ising models). W. N. YESSEN The 1D quasi-periodic quantum Ising chain
For a general overview of quasi-periodic (including Fibonacci) Ising models, see,for example, [29].2.1.
The model.
Let J , J >
0. Construct a { J , J } -valued sequence (cid:110) (cid:101) J n (cid:111) n ∈ N by applying repeatedly the Fibonacci substitution rule on two letters: J (cid:55)→ J J and J (cid:55)→ J , (1)starting with J : J (cid:55)→ J J (cid:55)→ J J J (cid:55)→ J J J J J (cid:55)→ · · · , at each step substituting for J and J according to the substitution rule (1). Bythis procedure an infinite sequence is constructed, which we call (cid:110) (cid:101) J n (cid:111) n ∈ N (see [59]for more details on substitution sequences).Let (cid:98) J k be the finite word after k applications of the substitution rule. It is easyto see that the following recurrence relation holds: (cid:98) J k +1 = (cid:98) J k (cid:98) J k − . (2)The word (cid:98) J k has length F k , where F k is the k th Fibonacci number. The quasi-periodic (Fibonacci in our case) one-dimensional quantum Ising model on the finiteone-dimensional lattice of N nodes with transversal external field is given by the Ising Hamiltonian H = − N (cid:88) n =1 (cid:101) J n σ ( x ) n σ ( x ) n +1 − h N (cid:88) n =1 σ ( z ) n , where h > σ ( x ) , ( z ) j are spin-1 / x and z directions, respectively,given by σ ( x ) , ( z ) n = I ⊕ · · · ⊕ I (cid:124) (cid:123)(cid:122) (cid:125) n − ⊕ σ ( x ) , ( z ) ⊕ I ⊕ · · · ⊕ I (cid:124) (cid:123)(cid:122) (cid:125) N − n +1 times , where I is the 2 × σ ( x ) , ( z ) are the Pauli matrices given by σ ( x ) = (cid:18) (cid:19) and σ ( z ) = (cid:18) − (cid:19) . (3)The magnetic field h can be absorbed into interaction strength couplings J , J , so H can be rewritten as H = − N (cid:88) n =1 (cid:16) (cid:101) J n σ ( x ) n σ ( x ) n +1 + σ ( z ) n (cid:17) . (4)The Hamiltonian H in (4) acts on C N , where we assume periodic boundary con-ditions: σ ( x ) , ( z ) N +1 = σ ( x ) , ( z )0 . Here σ ( α ) j , α ∈ { x, z } , acts on the finite sequence( c , . . . , c N ) ∈ C N by acting by σ ( α ) from (3) on the j -th entry, while leaving theother entries unchanged. PECTRUM OF QUANTUM ISING QUASICRYSTAL 5
Fermionic representation.
The spin model from the previous section canin fact be attacked as a so-called free-fermion model by performing the so-calledJordan-Wigner transformation, that transforms the Pauli operators σ ( x ) , ( y ) intoanti-commuting Fermi creation and annihilation operators. This technique datesat least as far back as the classical paper by P. Jordan and E. Wigner on sec-ond quantization [43]. The advantage in performing this transformation, is thatthe resulting Hamiltonian, in terms of the Fermi operators, can be diagonalizeddue to the anticommuting property of Fermi operators (commonly known in thephysics community as the canonical commutation relations, or CCR for short). Fur-thermore, the Jordan-Wigner transformation is canonical (to conform to standardphysics terminology) in the sense that it is invertible (in particular, no informationis introduced and no information is lost by performing this transformation).For convenience, let us denote by H k the Hamiltonian in (4) on a lattice of size F k .We can then extend H k periodically to a Hamiltonian (cid:98) H k over the periodic infinitelattice with the unit cell of length F k . The operator H k , and hence also (cid:98) H k , can becast into fermionic representation by means of the Jordan-Wigner transformation: H k = (cid:88) i,j (cid:20) c † i A ij c j + 12 (cid:16) c † i B ij c † j + ( c † i B ij c † j ) † (cid:17)(cid:21) , (5)where c i , 1 ≤ i ≤ F k , are anticommuting fermionic operators and a † denotesHermitian conjugation of a . The terms A ij and B ij that appear in (5) are entriesof the matrices A, B given by A ii = − , A i,i +1 = A i +1 ,i = − (cid:101) J i , A ,F k = − (cid:101) J F k ; B i,i +1 = − (cid:101) J i , B i +1 ,i = (cid:101) J i , B ,F k = (cid:101) J F k ;all other entries being zero. This method is due to E. Lieb et. al. [51], and itsspecialization to the Hamiltonian (4) is presented in some detail in [24]. We donot go into any further details here and invite the reader to consult the mentionedworks.Now, after we have performed the Jordan-Wigner transformation, the energyspectrum of the Hamiltonian in (5) can be computed by solving the so-called c -numeric equation for λ :( A − B ) φ = λψ, (6) ( A + B ) ψ = λφ (see [51, Section B]). This equation can be written in the form [6, 13, 84]Φ i +1 = M i ( λ )Φ i , where Φ i = ( ψ i , φ i ) T , and M i ( λ ) = (cid:32) − / (cid:101) J i λ/ (cid:101) J i − λ/ (cid:101) J i ( λ − (cid:101) J i ) / (cid:101) J i (cid:33) . Thus the wave function Φ at site N is given byΦ N +1 = M N ( λ ) × M N − ( λ ) × · · · × M ( λ )Φ . W. N. YESSEN
Letting (cid:99) M k ( λ ) denote the transfer matrix over F k sites, using the recurrence relationin (2), we obtain (cid:99) M k +1 ( λ ) = (cid:99) M k ( λ ) × (cid:99) M k − ( λ ) , (7)for k ≥ (cid:98) H k , we see that the wave function at site nF k isgiven by Φ nF k = (cid:99) M k ( λ ) n Φ . The wave function over the infinite lattice should not diverge exponentially. Hencewe allow only those values of λ for which the eigenvalues of (cid:99) M k ( λ ) lie in [ − , (cid:99) M k ( λ ) is unimodular, this is equivalent to the requirement12 (cid:12)(cid:12)(cid:12) Tr (cid:99) M k ( λ ) (cid:12)(cid:12)(cid:12) ≤ . (8)Let x k ( λ ) = 12 Tr (cid:99) M k ( λ ) . Using the recursion relation (7), one may derive the recursion relation on the tracesgiven by (see [45, 48, 55] and, for a more general discussion, [62]) x k +1 = 2 x k x k − − x k − . Thus, in accordance with (8), we require that | x k | ≤
1. Define the so-called
Fi-bonacci trace map f : R → R by f ( x, y, z ) = (2 xy − z, x, y ) . (9)Set M − ( λ ) = (cid:18) J /J λ ( J − J ) / J J J /J (cid:19) ,M ( λ ) = (cid:18) − /J λ/ J − λ/ J ( λ − J ) / J (cid:19) ,M ( λ ) = (cid:18) − /J λ/ J − λ/ J ( λ − J ) / J (cid:19) = M ( λ ) × M − ( λ ) . Then x − ( λ ) = (cid:18) J J + J J (cid:19) / , x ( λ ) = λ − (4 + 4 J )8 J , x ( λ ) = λ − (4 + 4 J )8 J . It is convenient to absorb the factor 1 / λ and rewrite x ( λ ) = λ − (1 + J )2 J , x ( λ ) = λ − (1 + J )2 J . Define the line of initial conditions γ ( J ,J ) : ( −∞ , ∞ ) → R by γ ( J ,J ) ( λ ) = (cid:18) λ − (1 + J )2 J , λ − (1 + J )2 J , (cid:18) J J + J J (cid:19) / (cid:19) . (10)Let π denote projection onto the third coordinate, and define σ k ( J , J ) = (cid:8) λ : (cid:12)(cid:12) π ◦ f k ◦ γ ( J ,J ) ( λ ) (cid:12)(cid:12) ≤ (cid:9) , (11) PECTRUM OF QUANTUM ISING QUASICRYSTAL 7 where f k denotes k -fold composition f k = f ◦ f ◦ · · · ◦ f (cid:124) (cid:123)(cid:122) (cid:125) k times , k ≥ . For the sake of simplifying notation, we shall write simply σ k , keeping in mindits implicit dependence on the choice of J and J . With this setup, the set σ k isthe excitation spectrum of the periodic free-fermion model or, equivalently (via theinverse Jordan-Wigner transformation), the energy spectrum of the periodic spinmodel . We are interested in understanding the spectrum in the thermodynamiclimit, that is, k → ∞ .2.3. The problem and main results.
We wish to investigate the energy spec-trum of (cid:98) H k in the thermodynamic limit (that is, k → ∞ ). Since π ◦ f k ◦ γ ( J ,J ) ( λ )is a polynomial in λ , σ k is a union of finitely many compact intervals (in general,see the exposition on Floquet theory applicable to periodic Hamiltonians, in, forexample, [73, Chapter 4]). Supported by numerical evidence, it is believed that as k → ∞ , the sequence { σ k } k ∈ N shrinks to a Cantor set [7,13,24,84] (i.e. a nonempty,compact, totally disconnected set with no isolated points). In Theorem 2.1 belowwe make precise the notion of the energy spectrum in the thermodynamic limit,and we rigorously examine its multifractal nature. Before we continue, however, weneed to set up some notation.We denote the Hausdorff metric on P ( R ) by dist H : dist H ( A, B ) = max (cid:26) sup a ∈ A inf b ∈ B {| a − b |} , sup b ∈ B inf a ∈ A {| a − b |} (cid:27) . We denote the Hausdorff dimension of a set A by dim H ( A ), and the local Hausdorffdimension of A at a point a ∈ A by dim locH ( A, a ): dim locH ( A, a ) = lim (cid:15) → + dim H (( a − (cid:15), a + (cid:15) ) ∩ A ) . We are now ready to state the main result of this paper.
Theorem 2.1.
Fix J > . There exists r = r ( J ) ∈ (0 , such that for all J satisfying J /J ∈ (1 − r , r ) , J (cid:54) = J , the following statements hold.i. There exists a compact nonempty set B ∞ ( J , J ) ⊂ R such that σ k −−−−→ k →∞ B ∞ in the Hausdorff metric;ii. B ∞ ( J , J ) is a Cantor set;iii. dim locH ( B ∞ ( J , J ) , b ) depends continuously on b ∈ B ∞ ( J , J ) , is non-constantand is strictly between zero and one; consequently dim H ( B ∞ ( J , J )) ∈ (0 , ,and therefore the Lebesgue measure of B ∞ ( J , J ) is zero;iv. dim H ( B ∞ ( J , J )) is continuous in the parameters ( J , J ) . Convergence of the sequence { σ k } to a (nonempty, compact) limit and multi-fractal nature of this limit (statements (i) and (ii) of the theorem) was observednumerically in [24, 83, 84].We should add that we believe the restrictions on J , J in the statement ofTheorem 2.1 are not necessary; however, our present techniques do not extend tothe general case (i.e. to cover all values of J , J ). We record our belief here formallyas a conjecture: Conjecture 2.2.
The conclusion of Theorem 2.1 holds for all J , J > . W. N. YESSEN
We should remark here that the Ising Hamiltonian above is equivalent, via aunitary transformation, to the tight binding model:( T θ ) n = (cid:101) J n − θ n − + (1 + (cid:101) J n ) θ n + (cid:101) J n +1 θ n +1 . (12)In fact, it can be seen easily that solving (6) is equivalent to solving the equation( A + B )( A − B ) φ = λ φ. This, on the other hand, is equivalent to solving T φ = λ φ. Now, T is a member of a family of tridiagonal Hamiltonians investigated in [82](results of [82] were obtained some months after a preprint of the present paperappeared). From our results in [82], combined with Proposition 5.16 from Section5.3 below, it follows that with T from (12), we have σ ( T ) ⊂ R + and B ∞ = (cid:110) ±√ λ : λ ∈ σ ( T ) (cid:111) , where σ ( T ) is the spectrum of T .After B ∞ has been tied to the spectrum of T as above, results of [2] yield a proofof statement (i) of Theorem 2.1. Moreover, in this case the restrictions given in thestatement of Theorem 2.1 are not necessary; these restrictions are, however, stilla necessity to our techniques for proving the remaining statements (ii)–(iv). Sinceprevious numerical methods have relied entirely on the dynamics of the Fibonaccitrace map, motivated by providing a rigorous counterpart, we provide an alternativeproof of Theorem 2.1-i in Section 5.3, based entirely on dynamical properties of thetrace map.On another note, we have the following remark regarding the connection withthe Jacobi operator T . Remark . In [82] it is proved (using the techniques that we develop here) thatthe spectrum of T is a Cantor (in fact, multifractal) set of zero Lebesgue measure;moreover, we proved (heavily relying on the previous work for Schr¨odinger oper-ators) that the spectral measures are purely singular continuous. The followingquestion was justly raised by one of the referees of this article: What implicationswould pure singular continuity of the spectral measures have in the context of theIsing model?
While we have not investigated this question in detail, we postulate aconnection with spin-spin correlation decay, and point to [33] and references thereinfor some work in this direction.3.
Hierarchy of results
Given that the proof of Theorem 2.1 is quite technical, we provide below adiagram of lemmas, propositions and theorems, demonstrating their hierarchy, inhopes of making navigation through the technical passages of the paper easier ( T stands for Theorem, P stands for Proposition, L stands for Lemma, and R standsfor Remark). PECTRUM OF QUANTUM ISING QUASICRYSTAL 9 T T P P P L L L L R P P T T T P P L P P P P P L L C C P L L L L L L L L L Dynamics of the Fibonacci trace map
In the proof of Theorem 2.1 we shall employ dynamical properties of the Fi-bonacci trace map f , which we discuss in this section. For a brief overview ofbasic notions and notation from the theory of hyperbolic and partially hyperbolicdynamical systems, see Appendix B.2. We shall make more specific references tothe appendix throughout the text, as necessary.For convenience henceforth we shall refer to the sequence (cid:8) f k ( x ) (cid:9) k ∈ N as the positive , or forward, semiorbit of x , and denote it by O + f ( x ). The negative , or backward, semiorbit for k ∈ Z < , and the full orbit , for k ∈ Z , are defined similarlyand denoted, respectively, by O − f and O f = O + f ∪ O − f .Define the so-called Fricke-Vogt character [26, 27, 78] I ( x, y, z ) = x + y + z − xyz − . (13)Consider the family of cubic surfaces { S V } V ≥ given by S V = (cid:8) x ∈ R : I ( x ) = V (cid:9) . (14)For all V > S V is a smooth, connected 2-dimensional submanifold of R withoutboundary. For V = 0, S V has four conic singularities, P = (1 , , , P = ( − , − , , P = (1 , − , − , P = ( − , , − , (15) (a) V = 0 . V = 0 . V = 0 .
05 (d) V = 1 Figure 1.
Invariant surfaces S V for four values of V .away from which the surface is smooth (see Figure 1).One can easily check that f preserves the Fricke-Vogt character by verifying that f ( I ( x )) = I ( x ); hence f also preserves the surfaces { S V } (i.e. f : S V → S V ). Forconvenience we shall write f V for f | S V . In fact, since f is invertible with the inverse f − ( x, y, z ) = ( y, z, yz − x ), f : S V → S V is an analytic diffeomorphism.Since those points whose positive semiorbit is bounded play a crucial role inour analysis, it is convenient, for future reference, to state as a separate result thefollowing necessary and sufficient conditions for a semiorbit to be bounded. Proposition 4.1.
Let x k = π ◦ f k ( x , x , x − ) . We have the following.(1) Assume | x − | ≤ C for some C ≥ . The sequence { x k } k ≥− is unboundedif and only if there exists k ≥ such that | x k − | ≤ C and | x k | , | x k +1 | > C. PECTRUM OF QUANTUM ISING QUASICRYSTAL 11 (2) A sufficient condition for { x k } k ≥− to be unbounded is that there exists k ≥ such that | x k | , | x k +1 | > and | x k | | x k +1 | > | x k − | . Remark . As defined earlier, π denotes projection onto the third coordinate. Proof.
For the proof of (1), see, for example, [15, Proposition 5.2] (replace 1 with C ). For (2), see [46, 48] ((2) also follows from the aforementioned proof of (1)). (cid:3) Remark . For detailed analysis of orbits of trace maps, see [61].Another result that we conveniently state as a separate statement establishesthat the set of all points with bounded positive semiorbit is a closed set. This is adirect consequence of Proposition 4.1.
Lemma 4.4.
For V ≥ and ∞ ≥ V ≥ V , the set of all points of (cid:83) V ∈ [ V ,V ] S V whose forward semiorbit is bounded is a closed set.Proof. If O + f ( x , x , x − ) is unbounded, then there exists a k ≥ f k ( x , x , x − ) satisfies (1) of Proposition 4.1, and hence also satisfies (2),which is an open condition. (cid:3) Another consequence of Proposition 4.1 that will be used later is Proposition 4.5below; for a proof see [15, Proposition 5.2], or (in a more general context) [61].
Proposition 4.5.
The positive semiorbit is unbounded if and only if (cid:8) f k ( x ) (cid:9) k ∈ N escapes to infinity in every coordinate. Most conclusions about the dynamics of f that we shall derive and use comefrom the knowledge of dynamics of f on the surfaces S V , i.e. dynamics of f V (notsurprisingly, since these surfaces are invariant under f ). In the following sectionswe recall some known results about dynamics of f V , V ≥
0, as well as prove somenew results necessary for the present investigation.4.1.
Dynamics of f V for V ≥ . In this and the following sections, we shall usethe notation and terminology from Appendix B.4.1.1.
Hyperbolicity of f V for V > . The following result on hyperbolicity of theFibonacci trace map will serve as the main tool for us. For definition of a locallymaximal transitive hyperbolic set (and the notion of (1 ,
1) splitting) see SectionB.1.
Theorem 4.6 (M. Casdagli in [12], D. Damanik and A. Gorodetski in [18], and S.Cantat in [11] ) . For
V > let Ω V = { p ∈ S V : O f ( p ) is bounded } . Then Ω V is a Cantor set, it is f V -invariant compact locally maximal transitivehyperbolic set in S V (with (1 , splitting). Moreover, Ω V is precisely the set ofnonwandering points of f V (a point p is nonwandering if for any neighborhood U of p and N ∈ N , there exists n ≥ N such that f n ( U ) ∩ U (cid:54) = ∅ ). The special case of V ≥
64 was done by M. Casdagli in [12]. D. Damanik and A. Gorodetskiextended the result to all
V >
V >
Remark . It follows that for
V >
0, for any x ∈ Ω V , W sloc ( x ) ∩ Ω V and W uloc ( x ) ∩ Ω V are Cantor sets ( W sloc ( x ) denotes the local stable manifold at x , and W uloc ( x )denotes the local unstable manifold at x – see Appendix B.1.2 for definitions andproperties of these objects). Remark . In fact, f V satisfied what is called Smale’s Axiom A [68]. The gen-eral theory of Axiom A diffeomorphisms is extensive and forms one of the centralthemes in the modern theory of dynamical systems. Suffice it to say that Axiom Adiffeomorphisms are the hallmark of chaotic dynamics.The preceding theorem is of importance, because as a consequence of it, the setof points whose forward semiorbit is bounded can be endowed with a sensible geo-metric structure, which is the subject of Corollary 4.9 below. The corollary followsfrom general principles in hyperbolic dynamics (and has been employed implicitlyin a number of previous works, especially [12,18]; for the technical details, see Corol-lary 2.5 and the discussion preceding it in [21]. In the statement of the corollary, W s (Ω V ) stands for (cid:83) x ∈ Ω V W s ( x ), where W s ( x ) is the global stable manifold (incontrast to the local one from Remark 4.7 above). For details, see equations (49)and (50) in Section B.1.2. Corollary 4.9.
For
V > , x ∈ S V , O + f ( x ) is bounded if and only if x ∈ W s (Ω V ) . Dynamics of f V for V = 0 . As mentioned above, the surface S is smootheverywhere except for the four singularities P , . . . , P (see (15) and Figure 1). Letus set, and henceforth fix, the following notation S = { ( x, y, z ) ∈ S : | x | , | y | , | z | ≤ } . (16)It is easily seen that S is invariant under f . Moreover, f | S is a factor of thehyperbolic diffeomorphism on T , A ( θ, φ ) = (( θ + φ ) , θ )(mod 1) , (17)via the factor map F : ( θ, φ ) (cid:55)→ (cos 2 π ( θ + φ ) , cos 2 πθ, cos 2 πφ ) . (18)By a factor we mean f | S ◦ F = F ◦ A . The map F is not, however, a conjugacy in the sense of (48) from Section B.1.2,since F is a two-to-one map. The dynamics of f on { P , . . . , P } is as follows. f : P (cid:55)→ P ; f : P (cid:55)→ P (cid:55)→ P (cid:55)→ P . It is easy to see that A : T → T is a hyperbolic map with the full space T constituting a compact hyperbolic set (such systems are more commonly known as Anosov diffeomorphisms ). Even though F : A → S is not a conjugacy, the behav-ior of Df on the tangent bundle of S away from the singularities P , . . . , P stillinherits the hyperbolic behavior from D A via DF (this is stated more precisely inLemma 5.3 below). However, the singularities (or, better to say, dynamics near thesingularities) requires special treatment. We now concentrate on a neighborhood of P ; similar results hold for the other singularities due to the symmetries of f (seeSection A.2 for details). PECTRUM OF QUANTUM ISING QUASICRYSTAL 13
Figure 2.
Per ( f ) in a neighborhood of P .Let Per ( f ) denote the set of period-two periodic points for f (note that Per ( f )is precisely what is called ρ in Section A.1). A direct computation shows thatPer ( f ) = (cid:26) ( x, y, z ) : x ∈ (cid:18) −∞ , (cid:19) ∪ (cid:18) , ∞ (cid:19) , y = x x − , z = x (cid:27) . Let ϑ ( x ) = (cid:18) x, x x − , x (cid:19) , ϑ : (cid:18) −∞ , (cid:19) ∪ (cid:18) , ∞ (cid:19) → R (19)be the curve of these periodic points (see Figure 2). We have Also, I ( ϑ ( x )) ≥ , (20)with I ( ϑ ( x )) = 0 if and only if x = 1, where ϑ (1) = P . On the other hand, Df P = −
11 0 00 1 0 which is similar to −√ − √ . Hence T P R splits as T P R = E s P ⊕ E cP ⊕ E u P , (21)where E sP corresponds to the subspace spanned by the eigenvector correspondingto the largest eigenvalue of Df (which is strictly larger than one), E uP correspondsto the subspace spanned by the eigenvector of Df P corresponding to the smallesteigenvalue of Df (which is the reciprocal of the largest one), and E cP correspondsto the subspace spanned by the eigenvector of Df P corresponding to the eigenvalue1, which is also the tangent space of Per ( f ) at the point P (see Section A.1 fora slightly more detailed discussion). Now using invariance of Per ( f ) under f , andcombining (20) with Theorem 4.6 for x (cid:54) = 1 and (21) for x = 1, we get that the curve ϑ is normally hyperbolic in a neighborhood of P (for definitions and properties,see Section B.3). Hence we can apply Theorem B.2 and, in the notation of SectionB.3 (see equation (54)) , we get that W csloc (Per ( f )) and W culoc (Per ( f )) are smoothtwo-dimensional submanifolds of R (roughly speaking, the manifold W csloc (Per ( f ))is precisely the set of those points of R which lie in some (cid:15) -neighborhood of Per ( f ) and whose positive semiorbit does not leave this neighborhood, and similarly for W culoc (Per ( f )) with f replaced by f − ; more details are given in Section B.3).Moreover, by invariance of the surfaces S V under f , it follows that W cs , culoc (Per ( f )) ∩ S V is precisely the one-dimensional manifold W s , uloc (Per ( f ) ∩ S V ) on S V (see SectionsB.1.2 and B.3), for V >
0. When V = 0, W cs , culoc (Per ( f )) ∩ S \{ P } forms the localstrong-stable/unstable one-dimensional submanifold (as defined in equation (52) ofSection B.3 ) on S consisting of two smooth branches that connect smoothly (whenviewed as submanifolds of R ) at P . Points of the strong-stable manifold convergeto P under iterations of f ; the same happens under iterations of f − for pointson the strong-unstable manifold. The strong-stable and strong-unstable manifoldsintersect at P .We have d ( I ◦ ϑ ) dx (cid:54) = 0for all x (cid:54) = 1. Hence ϑ , and therefore also W cs , culoc (Per ( f )), intersect S V trans-versely for V >
0. On the other hand, W cs , culoc (Per ( f )) has quadratic tangencywith S along the strong-stable/unstable submanifold (see also [18, Section 4]).This is a crucial point to make, since the proof of Theorem 2.1 relies heavily ontransversality of intersection of the line of initial conditions, γ ( J ,J ) , from (10),with W cs (Per ( f )) (which is defined below). In order to prove transversality, weproceed by perturbation analysis starting with γ ( J ,J ) with J = J . In this case γ lies on S , and since W cs (Per ( f )) is tangent to S , γ is tangent to W cs (Per ( f ))(this shows in particular that proving transversality for J (cid:54) = J is not a trivialtask). We then show that if J /J is close to 1, and not equal to 1, γ ( J ,J ) istransversal to W cs (Per ( f )). The situation is further complicated by the fact thatwe do not have any explicit, or quantitative, knowledge about W cs (Per ( f )), andmust rely only on qualitative analysis.The manifolds W csloc (Per ( f )) and W culoc (Per( f )) can be extended globally to W cs (Per ( f )) = (cid:91) n ∈ N f − n ( W csloc (Per ( f )))and W cu (Per ( f )) = (cid:91) n ∈ N f n ( W culoc (Per ( f ))) . In this case W s , u (Per ( f ) ∩ S V ) = W cs , cu (Per ( f )) ∩ S V for V >
0. For V = 0,these form branches of global strong-stable and strong-unstable submanifolds of S that connect at P - these branches are injectively immersed one-dimensionalsubmanifolds of S .Similar results hold for P , P , P . Indeed, as V takes on positive values, thepoints P , P and P bifurcate from three cycles to six cycles. These six cycles formthree smooth curves, one through each P , P and P . Considering f in place of f ,it is easy to show, as in the case of P above, that each curve is normally hyperbolic.For further details, refer to Section A.1.Let us fix the following notation. Denote by W s , u i the center-stable/unstable2-dimensional invariant manifold to the normally hyperbolic curve through P i . Wedenote by W s , u i, loc a small neighborhood of the normally hyperbolic curve in W s , u i . Asan aside, we should mention that in fact the normally hyperbolic curve is precisely W si, loc ∩ W ui, loc , provided that the neighborhood W s,ui, loc is taken sufficiently small, PECTRUM OF QUANTUM ISING QUASICRYSTAL 15 and this intersection is transversal (we should note that there exist, in fact infinitelymany , other intersections between W si and W ui ).In particular, the orbit O + f ( x ) (respectively, O − f ( x )), for x ∈ S , is boundedif and only if either x ∈ S or x ∈ W s i ∩ S (respectively, x ∈ W u i ∩ S ) for some i ∈ { , . . . , } .4.2. Partial hyperbolicity: center-stable and center-unstable manifolds.
In the previous section we proved normal hyperbolicity of f on certain submanifolds(namely, the curves of periodic points through the singularities) and derived theexistence of two-dimensional analogs of stable and unstable manifolds. The resultof the previous section is a special case of a more general fact, which is the subjectof Proposition 4.10. For the definition of a partially hyperbolic set (as well as thenotion of (1 , ,
1) splitting), see Section B.2. Note also that the results of theprevious section cannot be encapsulated into the following proposition, because inwhat follows, the results are stated for surfaces S V , V >
0; due to presence ofsingularities P , . . . , P in S , we had to treat the case of S separately. Proposition 4.10.
Let M = (cid:91) V ∈ (0 , ∞ ) S V . For V ≥ V > let Ω = (cid:91) V ∈ [ V ,V ] Ω V . Then M is a smooth, connected, f -invariant 3-dimensional submanifold of R , Ω ⊂ M is compact, f -invariant partially hyperbolic set with (1 , , splitting, andthe following holds.There exist two families, denoted by W s and W u , of smooth 2-dimensional con-nected manifolds injectively immersed in M , whose members we denote by, respec-tively, W cs and W cu , and call center-stable and center-unstable manifolds, with thefollowing properties.(1) The family W s , u is f -invariant;(2) For every x ∈ Ω there exist unique W cs ( x ) ∈ W s and W cu ( x ) ∈ W u suchthat x ∈ W cs ( x ) ∩ W cu ( x ) ;(3) Conversely, for every W cs ∈ W s and W cu ∈ W u , there exist x, y ∈ Ω suchthat x ∈ W cs and y ∈ W cu . In fact, if W cs1 , W cs2 ∈ W s and W cu1 , W cu2 ∈ W u with W cs1 ∩ W cs2 (cid:54) = ∅ and W cu1 ∩ W cu2 (cid:54) = ∅ , then W cs1 = W cs2 = W cs ( x ) forsome x ∈ Ω , and W cu1 = W cu2 = W cu ( y ) for some y ∈ Ω .(4) For any V > and any W cs , cu ∈ W s , u , W cs , cu ∩ S V = W s , u ( x ) for every x ∈ Ω V ∩ W cs , cu ; moreover, this intersection is transversal.Proof. All statements about M , as well as compactness of Ω, are trivially true.The surfaces { S V } , V >
0, are all diffeomorphic. Fix (cid:101)
V >
0, and let π V : S (cid:101) V → S V be a diffeomorphism, depending smoothly on V , with π (cid:101) V = Id | S (cid:101) V . Now, after asmooth change of coordinates, f : M → M may be considered as a skew productof identity on an interval I with a map on S (cid:101) V : I × S (cid:101) V I × S (cid:101) V (cid:83) V S V (cid:83) V S V (cid:101) π G f (cid:101) π where (cid:101) π ( V, x ) = π V ( x ), and G ( V, x ) = ( V, ˜ π − ◦ f ◦ ˜ π ( V, x )). Now all statementsabout Ω follow from Theorem 4.6 (in particular, noting that Ω (cid:101) V is hyperbolic).We now construct the family W s . The family W u can be constructed similarlyby considering f − in place of f .Fix (cid:101) V > x ∈ Ω (cid:101) V . Take δ > (cid:101) V − δ >
0) andfor V ∈ [ (cid:101) V − δ, (cid:101) V + δ ] let H V : Ω (cid:101) V → Ω V be the topological conjugacy (see SectionB.1.1). Then W csloc ( x ) = (cid:83) V ∈ ( (cid:101) V − δ, (cid:101) V + δ ) W sloc ( H V ( x )) is a smooth two-dimensionalmanifold (see [36, Section 6] for proof of smoothness) that can be extended to all V , and the sought W cs ( x ) is then given by W cs ( x ) = (cid:91) n ∈ N f − n ( W csloc ( x )) . The collection { W cs ( x ) } x ∈ Ω (cid:101) V gives the family W s .Fix x ∈ Ω and consider the curve V (cid:55)→ H V ( x ) in a neighborhood of x . Thiscurve is the intersection of W csloc ( x ) with W culoc ( x ), hence is smooth. Since f V dependssmoothly, and hence Lipschitz-continuously (when restricted to a compact subset),on V , by [34, Theorem 7.3], H V is the fixed point of a contracting map on a certainBanach space that depends Lipschitz-continuously on V . So by [34, Theorem 1.1],if V > C > V close to V , (cid:107) H V ( x ) − H V ( x ) (cid:107)| V − V | ≤ C, proving transversality of intersection of { H V ( x ) } V with S V . Hence W cs , cu ( x ) in-tersects S V , V >
0, transversely, as claimed. (cid:3)
Remark . From the proof of Proposition 4.10 and Section B.1.2 it is evident thatthe local center-stable and center-unstable manifolds at x , W csloc ( x ) and W culoc ( x ), de-pend continuously on the point x in the C topology. Consequently, by compactnesswe obtain uniform (in x ) transversality of W cs , culoc ( x ) with the surface S V , V >
Proposition 4.12.
Let M be as in Proposition 4.10. Suppose γ : [0 , → M issmooth and regular. Suppose further that γ intersects the members of W s trans-versely with its endpoints not lying on center-stable manifolds. Then the set of allpoints x ∈ [0 , for which O + f ( f ◦ γ ( x )) is bounded is a Cantor set.Proof. By construction of the center-stable manifolds, for s ∈ [0 , γ ( s ) has abounded forward orbit if and only if γ ( s ) belongs to a center-stable manifold.Compactness follows from Lemma 4.4. Now absence of isolated points and totaldisconnectedness follow from Remark 4.7 and Proposition 4.10. (cid:3) We are ready to prove Theorem 2.1.
PECTRUM OF QUANTUM ISING QUASICRYSTAL 17 Proof of main results
In this section we prove Theorem 2.1. The proof of the theorem is inherentlytechnical and involves nontrivial notions from the theory of hyperbolic and par-tially hyperbolic dynamical systems. To make the reading easier where possible,we have included as a separate section the necessary notions from the theory ofdynamical systems in Appendix B. Throughout this section, references are madeto the appendix where appropriate. The reader is also advised to use Section 3 asa road map.5.1.
A short comparison and contrast with Schr¨odinger operators.
Thosereaders who are familiar with the results and methods of spectral theory of dis-crete one-dimensional quasi-periodic Schr¨odinger operators may justly inquire asto why we couldn’t simply apply the methods that have already been developedfor Schr¨odinger operators (see [16] for a broad overview). To answer this question,let us quickly recall the setup for the Schr¨odinger operator, denoted by H . Theoperator acts on (cid:96) ( Z ) as( Hφ ) n = φ n − + φ n +1 + V ω n φ n , where V ∈ R , and ω is the sequence obtained by performing the Fibonacci substi-tution on the letters 0 and 1, starting with, say, 0: 0 (cid:55)→ (cid:55)→ (cid:55)→ (cid:55)→ · · · .This sequence can then be naturally extended to the left. For details, see the re-cent survey [16] and references therein. It turns out that there exists γ : R → R ,explicitly given by γ ( E ) = (cid:18) E − V , E , (cid:19) , such that E belongs to the spectrum of H if and only if O + f ( γ ( E )) is bounded.On the other hand, we have I ( γ ( E )) = V /
4, which is clearly independent of E and is non-negative (and is zero if and only if V = 0, which corresponds to the free Laplacian case, whose spectrum is [ − , f needs to beconsidered only on one surfaces, S V / , for any chosen and fixed V > I depends on the spectral parameter,forcing us to consider the action of f on all S V ≥ at once. It turns out thatthis is also what is responsible for multifractality (i.e. nonuniform local scalingproperties) of the spectrum (in contrast to the case of Schr¨odinger operators). Letus conclude by mentioning that we have encountered the same difficulties (with thesame consequences of multifractality) in a few other models ( see [21, 80, 82]).5.2. Preliminary technical platform.
The appearance of λ in γ ( J ,J ) in (10)makes γ symmetric in λ with respect to the origin. By abuse of notation, let uswrite λ in place of λ , where λ is allowed to take values in [0 , ∞ ).Take r = J /J and, for convenience, let us also write γ r in place of γ ( J ,J ) .Hence γ r ( λ ) = (cid:18) λ − (1 + J )2 J , λ − (1 + r J )2 rJ , r r (cid:19) . (22) Proposition 5.1.
For every J > , there exists r = r ( J ) ∈ (0 , , such that forall r ∈ (1 − r , r ) and r (cid:54) = 1 , the curve γ r in (22) intersects the center-stablemanifolds transversely.Proof. We begin by showing that γ r intersects uniformly transversely the manifolds W s i , i ∈ { , . . . , } . We shall see later (actually, we won’t prove this but point tothe work of S. Cantat [11]) that W si , for i ∈ { , . . . , } , form a dense (in the C topology) subfamily of the family W s from Proposition 4.10, and the conclusionof Proposition 5.1 will follow. We shall then combine this result with Proposition4.12. Lemma 5.2.
For all r sufficiently close to one and not equal to one, γ r intersects W s i , for i ∈ { , . . . , } , uniformly transversely.Proof. Returning to the map A in (17), we see that A is hyperbolic and is givenby the matrix A = ( ) with eigenvalues µ = 1 + √
52 and − µ − = 1 − √ . Let us denote by v s , v u ∈ R the stable and unstable eigenvectors of A : A v s = − µ − v s , A v u = µ v u , (cid:107) v s , u (cid:107) = 1 . Fix some small ζ > ζ <
1) and define the stable and unstablecone fields on R in the following way: K s ( p ) = (cid:8) v ∈ T p R : v = v u v u + v s v s , | v s | ≥ ζ − | v u | (cid:9) , (23) K u ( p ) = (cid:8) v ∈ T p R : v = v u v u + v s v s , | v u | ≥ ζ − | v s | (cid:9) . These cone fields are invariant: A ( K u p ) ⊂ Int( K u ( Ap )) and A − ( K s ( p )) ⊂ Int( K s ( A − p )) . Also, the iterates of the map A expand vectors from the unstable cones, and theiterates of the map A − expand vectors from the stable cones. That is, there existsa constant C > ∀ v ∈ K u ( p ) ∀ n ∈ N (cid:107) A n v (cid:107) > Cµ n (cid:107) v (cid:107) , ∀ v ∈ K s ( p ) ∀ n ∈ N (cid:13)(cid:13) A − n v (cid:13)(cid:13) > Cµ n (cid:107) v (cid:107) . The families of cones { K s ( p ) } p ∈ R and { K u ( p ) } p ∈ R can also be considered on T = R / Z .The differential of the semiconjugacy F in (18), DF , sends these cone families to Df -invariant stable and unstable cone families on S \ { P , . . . , P } . Let us denotethese images by {K s } and {K u } , respectively. It is clear that away from a neighbor-hood of singularities, the cones K s,u have nonzero size. The following lemma satesthat the size of these cones is uniformly bounded away from zero on S \{ P , . . . , P } . Lemma 5.3 ([18, Lemma 3.1]) . The differential of the semiconjugacy DF inducesa map of the unit bundle of T to the unit bundle of S \{ P , . . . , P } . The derivativesof the restrictions of this map to a fiber are uniformly bounded. In particular, thesizes of cones in families {K s } and {K u } are uniformly bounded away from zero. PECTRUM OF QUANTUM ISING QUASICRYSTAL 19 W cs loc ( P i ) F i C u r v e o f p e r . p n t s . UP i S S V Figure 3.
Fix j ∈ { , . . . , } . Let α be a smooth curve in W sj that is C close to, anddisjoint from, the curve of periodic points through the singularity P j . Assume alsothat α ∩ S ∈ S . Let F j be the fundamental domain in W sj bounded by α and f − ( α ) (for the definition of a fundamental domain, see Section B.1.3). Let U bea neighborhood of { P , . . . , P } in R so small, that U ∩ F j = ∅ (see Figure 3).Given V > V ∈ [0 , V ), the surface S V \ U consists offive smooth connected components (with boundary), one of which is compact. Let S V,U denote the compact component. The family { S V,U } V ∈ [0 ,V ) depends smoothlyon V , that is, there exists a family of smooth projections, depending smoothly on V : { π V : S ,U → S V,U } V ∈ [0 ,V ) , π = Id S ,U . Assuming V is sufficiently small, Dπ V carries the cones {K s , u } to nonzero coneson S V,U ; denote these cones by {K s , u V } .In the next series of lemmas we shall construct in some special way some familiesof cone fields and show that they are invariant under the action by Df . The reasonfor doing this is the following. As we have already mentioned, we do not have anyquantitative information about W si . In particular, we cannot check directly whetherthe given line, γ r , is transversal to W si . However, based on qualitative propertiesof W si and quantitative information that we do have about the surfaces S V , V ≥ γ r , we can construct a cone field with the following properties. • This cone field is transversal to W si at points in the set F i ∩ S V for all V > W si with S ); • The line γ r falls inside this cone field for all r sufficiently close to 1 (we cancheck this directly from the explicit expression of γ ); • This cone field is invariant under the action by Df (we use both, the ge-ometry of S V and some dynamical properties of f ).We then take any point in γ r ∩ W si , say γ r ( x ), and iterate it by f until it fallsinside F i ; say f k ( γ r ( x )) ∈ F i . By the invariance of the constructed cone field, wemust then have Df k ( γ (cid:48) ( x )) inside the cone at f k ( x ), which is transversal to W si .Thus γ (cid:48) ( x ) must have been transversal to T γ ( x ) W si to begin with.Below, Lemmas 5.4, 5.5 and 5.6 establish invariance of {K s,uV } , as well as scalingproperties of vectors from these cones under the action by Df , by consideringdifferent cases. The final result is recorded in Corollary 5.7. Lemma 5.4.
For all N ∈ N there exists V > sufficiently small such that for all k ∈ Z ≥ with k ≤ N and all V ∈ [0 , V ) , if x ∈ S V,U and f k ( x ) ∈ S V,U , then Df kx ( K u V ( x )) ⊂ Int( K u V ( f k ( x ))) . Proof.
Since π V depends smoothly on V , for x ∈ S ,U , the cones K u ( x ) and K u V ( π V ( x )) are close provided that V is close to zero. Since f V depends smoothlyon V , Df kV and Df k are close. In particular, for a given x ∈ S ,U with f k ( x ) ∈ S ,U ,if π V ( x ) ∈ S V,U and f k ( π V ( x )) ∈ S V,U , then Df kV ( K u V ( π V ( x ))) and Df k ( K u ( x ))are close. Thus by compactness of the surfaces S V,U , we can choose V suitablysmall, so that the conclusion of the Lemma holds. (cid:3) Lemma 5.5 ([18, Lemma 5.4]) . Assuming U is sufficiently small, there existssufficiently large N ∈ N and sufficiently small V > , such that for all V ∈ [0 , V ) the following holds. If x ∈ S V,U and k is the smallest number such that f k ( x ) ∈ S V,U and k ≥ N , then Df kx ( K u V ( x )) ⊂ Int( K u V ( f k ( x ))) . Lemma 5.6 ([18, Lemma 5.2]) . There exists V > sufficiently small, C > and µ > , such that for all V ∈ [0 , V ) the following holds. If x ∈ S V,U and for k ∈ N , f k ( x ) ∈ S V,U , and if v ∈ K u V ( x ) , then (cid:13)(cid:13) Df kx ( v ) (cid:13)(cid:13) ≥ Cµ k (cid:107) v (cid:107) . Combination of Lemmas 5.4, 5.5, 5.6 gives
Corollary 5.7.
Assuming U is small, there exists V > sufficiently small, C > and µ > such that for all V ∈ [0 , V ) , the following holds. If x ∈ S V,U , v ∈ K u V ( x ) , k ∈ N and f k ( x ) ∈ S V,U , then Df kx ( K u V ( x )) ⊂ Int( K u V ( f k ( x ))) and (cid:13)(cid:13) Df kx ( v ) (cid:13)(cid:13) ≥ Cµ k (cid:107) v (cid:107) . With U and V satisfying the hypothesis of Corollary 5.7, let us construct thefollowing cone field on S V,U , for V ∈ [0 , V ) and η > K ηV ( x ) = (cid:110) ( u , v ) ∈ T x S V,U ⊕ ( T x S V,U ) ⊥ : u ∈ K u V ( x ), (cid:107) v (cid:107) ≤ η √ V (cid:107) u (cid:107) (cid:111) . (24)The cone field from (24) is of central importance, as it is the one that will beshown to be transversal to W si , and to contain the line γ r . However, as it is defined,it may not be invariant in the sense of the preceding corollary. The next lemmaestablishes that it is almost invariant (which is enough for us). PECTRUM OF QUANTUM ISING QUASICRYSTAL 21
Lemma 5.8.
For every (cid:98) η > there exists η = η ( (cid:98) η ) > , η < (cid:98) η , and V > sufficiently small, such that for any V ∈ [0 , V ) , any x ∈ S V,U , k ∈ Z ≥ , if f k ( x ) ∈ S V,U , then Df kx ( K ηV ( x )) ⊂ K (cid:98) ηV ( f k ( x )) . Proof.
Smooth dependence of the surfaces { S V } V > on V and invariance under f implies the following. Lemma 5.9.
For any
V > , x ∈ S V and k ∈ Z , if v ∈ ( T x S V ) ⊥ , then (cid:13)(cid:13)(cid:13) Proj ( T fk ( x ) S V ) ⊥ (cid:0) Df kx ( v ) (cid:1)(cid:13)(cid:13)(cid:13) = (cid:107)∇ I ( x ) (cid:107)(cid:107)∇ I ( f k ( x )) (cid:107) (cid:107) v (cid:107) , (25) where ∇ I is the gradient of the Fricke-Vogt character (see (13) ). In particular,there exists D > such that for all V ∈ (0 , V ) and any x ∈ S V,U , if f k ( x ) ∈ S V,U ,then for every v ∈ ( T x S V ) ⊥ , we have (cid:13)(cid:13)(cid:13) Proj ( T fk ( x ) S V,U ) ⊥ (cid:0) Df kx ( v ) (cid:1)(cid:13)(cid:13)(cid:13) ≤ D (cid:107) v (cid:107) . In fact, we can take D = sup (cid:26) (cid:107)∇ I ( x ) (cid:107)(cid:107)∇ I ( y ) (cid:107) : x, y ∈ S V,U , V ∈ [0 , V ] (cid:27) < ∞ . Proof.
Let M = (cid:91) V > S V . Integrate the gradient vector field x (cid:55)→ ∇ I ( x ) on M , and let α x denote a compactarc along the integral curve through x , say parameterized on [ − ,
1] with α x (0) = x .Let β = f k ( α x ). Then (cid:107)∇ I ( x ) (cid:107) = ( I ◦ α x ) (cid:48) (0) = ( I ◦ β ) (cid:48) (0) = C ∇ I ( f k ( x )) · ∇ I ( f k ( x )) , where C ∇ I ( f k ( x )) is the projection of β (cid:48) (0) onto ( T f k ( x ) S V ) ⊥ , C > (cid:13)(cid:13) C ∇ I ( f k ( x )) (cid:13)(cid:13) = (cid:107)∇ I ( x ) (cid:107)(cid:107)∇ I ( f k ( x )) (cid:107) · (cid:107)∇ I ( x ) (cid:107) . (cid:3) Let D be as in Lemma 5.9 and µ and C as in Corollary 5.7. Let k ∈ N bethe smallest number such that Cµ k > D . Fix N ∈ N with N > k . Let U ∗ be aneighborhood of { P , . . . , P } in R such that U ∗ ⊂ U , so small that if x ∈ U ∗ and l ∈ N is the smallest number such that f l ( x ) / ∈ U , then l > N . Case (i) . Suppose x ∈ S V,U , f k ( x ) ∈ S V,U and
N > k ≥ k . Then Cµ k > D ,hence the expansion in the cone K u V ( x ) dominates the expansion along the normal.On the other hand, the normal at x , under the action of Df kx , may tilt to the sideaway from K u V ( f k ( x )). However, since k < N , (cid:8) x, f ( x ) , . . . , f k ( x ) (cid:9) ⊂ S V,U ∗ . Bycompactness of S V,U ∗ , the angle between the image under Df kx of the normal at x and S V,U ∗ remains uniformly bounded away from zero. This, together with thefact that K u V ( x ) is mapped into the interior of K u V ( f k ( x )), allows us to choose V sufficiently small to compensate for the tilt in the normal. Hence for any η > V > Df kx ( K ηV ( x )) ⊂ K ηV ( f k ( x )). P i W u ( P i ) W s ( P i ) W c s ( P i ) W cu ( P i )Curve of per. pnts. Figure 4.
Case (ii) . If x ∈ S V,U , f k ( x ) ∈ F j ⊂ S V,U and k < N , then for sufficiently small η (depending only on N and independent of x ), K ηV ( x ) ⊂ K (cid:98) ηV ( f k ( x )); that is, giventhat the number of iterations does not exceed a given constant, the distortion canbe controlled. Case (iii) . We now handle the case when x , under iterates of f , passes through U ∗ . By symmetries of the map f , it is enough to consider only a neighborhood of P .Say U ∗ is a neighborhood of P . If U ∗ is sufficiently small, there exists a smoothchange of coordinates Φ : U ∗ → R such that Φ( P ) = (0 , ,
0) and the followingholds.Denote by W s , u1 ( P ) a small neighborhood of the point P on the manifold W s , u1 .We have • Φ(Per ( f )) is part of the line { x = 0 , z = 0 } ; • Φ( W s1 ( P )) is part of the plane { z = 0 } ; • Φ( W u1 ( P )) is part of the plane { x = 0 } ; • Φ( W s1 ∩ S ) is part of the line { y = 0 , z = 0 } ; • Φ( W u1 ∩ S ) is part of the line { x = 0 , y = 0 } .Denote S V = Φ( S V ). Then { S V } V > is a family of smooth surfaces dependingsmoothly on V , S is diffeomorphic to a cone, contains lines { y = 0 , z = 0 } and { x = 0 , y = 0 } , and at each nonzero point on those lines it has a quadratic tangencywith the xy - and zy -plane (see Figure 4).For a point p , denote its coordinates by ( x p , y p , z p ). Lemma 5.10 ([20, Propositions 3.12 and 3.13]) . Given C , C > , ρ > , thereexists δ > , C > , µ > and N ∈ N , such that for any δ ∈ (0 , δ ) , the followingholds. PECTRUM OF QUANTUM ISING QUASICRYSTAL 23
Let g : R → R be a C diffeomorphism such that(i) (cid:107) g (cid:107) C ≤ C ;(ii) The plane { z = 0 } is invariant under the iterates of g ;(iii) (cid:107) Dg ( p ) − A (cid:107) < δ for every p ∈ R , where A = ρ − ρ is a constant matrix.Introduce the following cone field on R : (cid:101) K p = (cid:26) v ∈ T p R , v = v xy + v z : | v z | ≥ C (cid:113) | z p | (cid:107) v xy (cid:107) (cid:27) . (26) Then for any p ∈ R satisfying | z p | ≤ ,(1) Dg ( (cid:101) K p ) ⊂ (cid:101) K g ( p ) ;(2) if (cid:12)(cid:12) z g N ( p ) (cid:12)(cid:12) > with N ≥ N , then for any v ∈ (cid:101) K p , if Dg N ( v ) = u xy + u z , then (cid:107) u xy (cid:107) < δ / | u z | and (cid:13)(cid:13) Dg N ( v ) (cid:13)(cid:13) ≥ Cµ N (cid:107) v (cid:107) . For a given δ >
0, if the neighborhood U ∗ of singularities is small enough, thenat every point p ∈ U ∗ the differential D (Φ ◦ f ◦ Φ − )( p ) satisfies condition (iii) ofLemma 5.10. Since the tangency of S with the plane { z = 0 } is quadratic, thereexists C > S from the cone D Φ( K u ) alsobelongs to the cone in (26). The same holds for vectors tangent to S V from thecones D Φ( K u V ) for V small enough. Therefore, Lemma 5.10 can be applied to allthose vectors. In particular, we have Lemma 5.11.
Assume U is so small that Lemma 5.10 can be applied. Thereexists C > such that if x ∈ S V,U , f ( x ) ∈ U and l ∈ N is the smallest numbersuch that f l − ( x ) ∈ U and f l ( x ) / ∈ U , then If l > N , then for any v ∈ T x R with D Φ( Df x ( v )) ∈ (cid:101) K Φ( f ( x )) , we have(1) Proj T fl ( x ) S V,U (cid:0) Df lx ( v ) (cid:1) ∈ K u V ( f l ( x )) ;(2) (cid:13)(cid:13) Df lx ( v ) (cid:13)(cid:13) ≥ Cµ l (cid:107) v (cid:107) , where µ is as in Lemma 5.10. Now, if U is sufficiently small and V > x ∈ S V,U with f ( x ) ∈ U , then for any η > D Φ( Df x ( K ηV ( x ))) ⊂ (cid:101) K Φ( f ( x )) . Hence Lemma 5.11 can be applied to vectors in Df ( K ηV ( x )). In particular, choosing U ∗ so small that N > N , we see that if x ∈ S V,U , f ( x ) ∈ U and k ∈ N is thesmallest number such that f k ( x ) / ∈ U , and (cid:8) x, f ( x ) , . . . , f k − ( x ) (cid:9) ∩ U ∗ (cid:54) = ∅ , then Df kx ( K ηV ( x )) ⊂ K ηV ( f k ( x )), for any η > η = (cid:98) η min (cid:26) CD , (cid:27) , where C is as in Corollary 5.7 and D is as in Lemma 5.9. (cid:3) We immediately obtain, as a consequence of Lemma 5.8, the following y xU U − − (cid:15) (cid:15) (1 , − , − γ r γ γ r S Figure 5. P P γ (0) γ S S V Figure 6.
Corollary 5.12.
Since the fundamental domain F j has quadratic tangency with S , there exists V > and (cid:98) η > such that for all V ∈ [0 , V ) and x ∈ S V,U ∩ F j ,the cone K (cid:98) ηV ( x ) is transversal to F j . Let η < (cid:98) η be as in Lemma 5.8. Then for all x ∈ S V,U ∩ W sj and v ∈ K ηV ( x ) , if x does not lie in the region bounded by α andthe curve of periodic points (i.e. x lies in (cid:83) n ≥ f − n ( F j )) , then v is transversal to W sj .Proof. For every x ∈ S V,U ∩ W sj that satisfies the hypothesis of the corollary, thereexists k ∈ N , such that f k ( x ) ∈ F j . The result follows by Lemma 5.8. (cid:3) Now recall the definition of γ r ( λ ) from (22). With I denoting the Fricke-Vogtcharacter (see (13)), we have I ( γ r ( λ )) = λ (cid:18) r − r (cid:19) ;(27) ∂I ( γ r ( λ )) ∂λ = 14 (cid:18) r − r (cid:19) . Consequently γ ( λ ) is the line that contains the singularities P and P (see (15))and lies entirely on S . For r (cid:54) = 1, γ r intersects the surfaces { S V } V ≥ transversely,intersecting each surface in a unique point, and intersects S when λ = 0. Case (i) (Assuming J (cid:54) = 1 ) . Assume J (cid:54) = 1. If λ = 0 then by (2) of Proposition4.1, O + f ( γ (0)) escapes. By continuity, there exists r ∈ (0 ,
1) such that for all r ∈ (1 − r , r ), the set (cid:110) x ∈ γ r : O + f ( x ) is bounded (cid:111) lies on a line segment Λ r whose endpoints belong to U (the neighborhood of sin-gularities) (see Figures 5 and 6). Let V be so small that Corollary 5.12 can beapplied, and r ∈ (0 ,
1) such that for all r ∈ (1 − r , r ), the line segment Λ r lies entirely in U ∪ (cid:16)(cid:83) V ∈ [0 ,V ) S V,U (cid:17) . Let η and (cid:98) η be as in Corollary 5.12. If λ > PECTRUM OF QUANTUM ISING QUASICRYSTAL 25 is such that γ r ( λ ) ∈ S V , from (27) we have ∂I ( γ r ( λ )) ∂λ = 1 λ I ( γ r ( λ )) . (28)Since the set { λ : γ r ( λ ) ∈ Λ r , r ∈ (1 − r , r ) } is bounded away from zero uni-formly in r , there exists C > λ , if γ r ( λ ) ∈ S V,U , then (cid:93) ( γ (cid:48) r ( λ ) , S V ) ≤ C V. (29)On the other hand, F − ( γ ∩ S ) = { θ = − φ } (see (18)), and (1 , −
1) is not aneigenvector of A , hence by taking K u ( p ) in (23) wider as necessary, we have thatfor all λ such that γ r ( λ ) ∈ S \ { P , . . . , P } , γ r ( λ ) ∈ Int( K u ). By continuity wehave, for all r sufficiently close to 1 and V close to 0, if λ is such that γ r ( λ ) ∈ S V,U ,then
Proj T γr ( λ ) S V ( γ r ) ∈ K u V . (30)Combined with (29), this gives γ r ( λ ) ∈ K ηV ( γ r ( λ )).Now suppose λ is such that γ r ( λ ) ∈ Λ r ∩ U . Let k ∈ N be the smallest numbersuch that f k ( γ r ( λ )) ∈ S V,U . If k < N , where N is as in Lemma 5.10, then γ r ( λ ) / ∈ U ∗ . The cones K u V defined on S V,U , V ∈ [0 , V ), can be defined on S V,U ∗ in the same way, by taking V smaller as necessary. Hence by taking r closer to 1as necessary, for all r ∈ (1 − r , r ), if λ is such that γ r ( λ ) ∈ S V,U ∗ , then (30)again holds. Assuming V was initially taken sufficiently small, we must have Df kγ r ( λ ) ( γ (cid:48) r ( λ )) ∈ K ηV ( f k ( γ r ( λ )) . Now suppose k > N . Lemma 5.13.
For all sufficiently small C -perturbations of γ , Φ( γ ) is tangentto the cones in (26) on Φ( U ) , with U a sufficiently small neighborhood of P .Proof. Observe that γ lies in the plane { z = 1 } . Notice, from (19), that the curveof periodic points, Per ( f ), is transversal to { z = 1 } at the point P . A simplecalculation shows that the eigenvector corresponding to the smallest eigenvalue of Df P is also transversal to { z = 1 } . Hence Φ( γ ) is transversal to W u ( P ) (a neigh-borhood of P in W u ), and so also to the plane { z = 0 } in the rectified coordinates.Hence all sufficiently small C -perturbations of Φ( γ ) are (uniformly) transversalto { z = 0 } , and therefore tangent to the cones in (26) in Φ( U ), for sufficiently small U . (cid:3) Now Lemma 5.11 can be applied. We get
Proj T fk ( γr ( λ )) S V (cid:16) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:17) ∈ K u V ( f k ( γ r ( λ )) . (31)On the other hand, by (29) we have (cid:13)(cid:13)(cid:13) Proj ( T γr ( λ ) S V ) ⊥ ( γ (cid:48) r ( λ )) (cid:13)(cid:13)(cid:13) ≤ C V (cid:107)∇ I ( γ r ( λ )) (cid:107) , and after applying (25) (Lemma 5.9) we obtain (cid:13)(cid:13)(cid:13) Proj ( T fk ( γr ( λ )) S V ) ⊥ (cid:16) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:17)(cid:13)(cid:13)(cid:13) =(32) = (cid:107)∇ I ( γ r ( λ )) (cid:107)(cid:107)∇ I ( f k ( γ r ( λ ))) (cid:107) (cid:13)(cid:13)(cid:13) Proj ( T γr ( λ ) S V ) ⊥ ( γ (cid:48) r ( λ )) (cid:13)(cid:13)(cid:13) ≤ C (cid:107)∇ I ( f k ( γ r ( λ ))) (cid:107) V. By compactness, for all V ∈ [0 , V ), x ∈ S V,U , (cid:107)∇ I ( x ) (cid:107) is uniformly bounded awayfrom zero. Thus, assuming V is sufficiently small, combining equations (31) and(32), we get Df kγ r ( λ ) ( γ (cid:48) r ( λ )) ∈ K ηV ( f k ( γ r ( λ ))) . We can now use Corollary 5.12 to conclude that Λ r intersects W s j transversely. Case (ii) (Assuming J = 1 ) . When J = 1, γ r ( λ ) has the form γ r ( λ ) = (cid:18) λ − , λ − (1 + r )2 r , r r (cid:19) . (33)Observe that if r (cid:54) = 1, then O + f ( γ r (0)) escapes, since f ( γ r (0)) satisfies (2) ofProposition 4.1. Hence for all r (cid:54) = 1 (and sufficiently close to 1) there exists λ ( r ) > λ < λ ( r ), O + f ( γ r ( λ )) escapes, by Lemma 4.4. We have Lemma 5.14.
There exists C > such that for all r (cid:54) = 1 sufficiently close to ,and λ ( r ) as above, if λ ≥ λ ( r ) and γ r ( λ ) ∈ S V , then Vλ ≤ C √ V .
Proof.
Let π : R → R denote projection onto the first coordinate. Since r ≈ λ ( r ) (cid:54) = 0 is close to zero, hence from (33), | π ( γ r ( λ )) | < . So by Proposition 4.1, O + f ( γ r ( λ )) will diverge if (cid:12)(cid:12) π ◦ f i ( γ r ( λ )) (cid:12)(cid:12) > i = 1 ,
2. Asimple calculation shows the existence of a constant
D >
0, independent of r , suchthat | π ◦ f ( γ r ( λ )) | ≤ or (cid:12)(cid:12) π ◦ f ( γ r ( λ )) (cid:12)(cid:12) ≤ ⇒ λ ≥ ( r − D .
Now, say γ r ( λ ) ∈ S V , and its forward orbit is bounded. Then λ ≥ λ ( r ). From(27), V = λ (cid:18) r − r (cid:19) . So √ Vλ = (cid:12)(cid:12) r − (cid:12)(cid:12) r √ λ ≤ (cid:12)(cid:12) r − (cid:12)(cid:12) r √ λ ≤ √ D ( r + 1)2 r . The right side above is uniformly bounded for all r away from zero. (cid:3) Suppose γ r ( λ ) ∈ S V , γ r ( λ ) ∈ U ∗ ∩ Λ r and k ∈ N is the smallest number such that f k ( γ r ( λ )) ∈ S V,U . Then λ (cid:54) = 0 and k > N . From equation (28) and a calculationsimilar to (32), it follows that (cid:13)(cid:13)(cid:13) Proj ( T fk ( γr ( λ )) S V ) ⊥ (cid:16) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:17)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) Proj ( T fk ( γr ( λ )) S V ) ⊥ (cid:16) Df kγ r ( λ ) ( Proj ( T γr ( λ ) S V ) ⊥ ( γ (cid:48) r ( λ )) (cid:17)(cid:13)(cid:13)(cid:13) (34) ≤ D λ V ≤ D C √ V ,
PECTRUM OF QUANTUM ISING QUASICRYSTAL 27 where C is as in Lemma 5.14 and D = sup (cid:107)∇ I ( x ) (cid:107) : x ∈ (cid:91) V ∈ [0 ,V ) S V,U . On the other hand, (cid:107) γ (cid:48) r ( λ ) (cid:107) > r . Now by Lemma 5.13, Lemma5.11 can be applied, so that by Part (2), with C > µ > (cid:13)(cid:13)(cid:13) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:13)(cid:13)(cid:13) ≥ Cµ k (cid:107) γ (cid:48) r ( λ ) (cid:107) > Cµ k . (35)Hence we have(1) Proj T fk ( γr ( λ )) S V (cid:16) Df kγ r ( λ ) ( γ (cid:48) r ( λ ) (cid:17) ∈ K u V ( f k ( γ r ( λ ))),(2) and from (34) and (35), (cid:13)(cid:13)(cid:13) Proj ( T fk ( γr ( λ )) S V ) ⊥ (cid:16) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:17)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Df kγ r ( λ ) ( γ (cid:48) r ( λ )) (cid:13)(cid:13)(cid:13) ≤ D C Cµ k √ V .
Hence (cid:93) ( Df kγ r ( λ ) ( γ (cid:48) r ( λ )) , S V ) ≤ (cid:101) η √ V , where (cid:101) η can be made arbitrarily small if k is sufficiently large (i.e. if U ∗ is initiallychosen sufficiently small). Hence Df kγ r ( λ ) ( γ (cid:48) r ( λ )) ∈ K ηV , and Corollary 5.12 can be applied. Case (iii) (If a point doesn’t enter F j ) . Assume that the point γ r ( λ ) lies onthe region of W s j bounded by the curve of periodic points and the curve α (seediscussion following Lemma 5.3). This is a finite region, hence contains at mostfinitely many points of intersection. So there exists r ∈ (0 , r ∈ (1 − r , r ), if γ r intersects this region, then this intersection is transversal.Combination of Cases (i), (ii) and (iii) gives transversality in Lemma 5.2. Toprove uniform transversality, observe that by Lemma 5.8, the cones { K ηV } aretransversal to { W s i } i ∈{ ,..., } for V > (cid:110) K η/ V (cid:111) are uniformly transversal to { W s i } i ∈{ ,..., } on surfaces away from S . Hence by taking r closer to one as necessary, but not equal to one, we can en-sure that γ r lies in the cones (cid:110) K η/ V (cid:111) , and hence intersects { W s i } i ∈{ ,..., } uniformlytransversely. This finally concludes the proof of Lemma 5.2. (cid:3) For justification of the following result, see [12, Section 2] and [11, Section 5.4].
Lemma 5.15.
For
V > , { W s i ∩ S V } i ∈{ ,..., } is dense (in the C topology) in thelamination W s (Ω V ) . Now from our construction of the family W s in Proposition 4.10, together withRemark 4.11, combined with Lemma 5.15, we get that W si ∈{ ,..., } is dense (in the C topology) in the family W s from Proposition 4.10. Proposition 5.1 now followsfrom Lemma 5.2. (cid:3) Proof of Theorem 2.1-(i).
The proof of Theorem 2.1-(i) relies entirely onthe dynamics of the trace map f , avoiding any spectral-theoretic considerations.We start with Proposition 5.16.
With γ r from (22) , define (cid:101) B ∞ ,r = (cid:110) λ ∈ [0 , ∞ ) : O + f ( γ r ( λ )) is bounded (cid:111) . Define also (cid:101) σ k,r = (cid:8) λ ∈ [0 , ∞ ) : (cid:12)(cid:12) π ◦ f k ( γ r ( λ )) (cid:12)(cid:12) ≤ (cid:9) , where π , as above, denotes projection onto the third coordinate. Then (cid:101) σ k −−−−→ k →∞ (cid:101) B ∞ in Hausdorff metric.Proof. For convenience we drop r and simply write (cid:101) B ∞ and (cid:101) σ k , keeping in mindimplicit dependence of these sets on r .We begin the proof with the following simple observation. Lemma 5.17.
For any
V > , if x = ( x , x , x ) ∈ Ω V , then one of the coordinatesof x is strictly smaller than one in absolute value.Proof. If all three coordinates are equal to one in absolute value, then x has − x is one of the foursingularities of S , but by hypothesis V > x cannot be an element of Ω V .If at least one but not all of its coordinates is equal to one in absolute value, andthe rest are strictly greater than one in absolute value, then by applying either f or f − = ( y, z, yz − x ), we shall obtain a point ( y , y , y ) with | y | , | y | > | y y | > | y | or | y | , | y | > | y y | > | y | . In the first case, by (2) of Proposition4.1, the point ( y , y , y ) has unbounded forward semiorbit; in the second case, bya similar result applied to f − , ( y , y , y ) has unbounded backward semiorbit. Ineither case, x cannot belong to Ω V . (cid:3) By construction of the center-stable manifolds, it follows that (cid:101) B ∞ is precisely theintersection of γ r with the center-stable manifolds. As in the proof of Corollary 5.12,this intersection occurs on a compact line segment Λ r along γ r . Now applicationof Lemma 4.4 shows that (cid:101) B ∞ is compact.By Lemma 5.17 it follows that for all λ ∈ (cid:101) B ∞ , there exists N λ ∈ N , such thatfor all k ≥ N λ , f k ( γ ( λ )) belongs to (cid:101) σ k ∪ (cid:101) σ k +1 ∪ (cid:101) σ k +2 . By compactness of (cid:101) B ∞ , thereexists such N ∈ N uniformly for all λ ∈ (cid:101) B ∞ . DefineΣ k = (cid:101) σ N + k ∪ (cid:101) σ N + k +1 ∪ (cid:101) σ N + k +2 . It is a simple observation that follows from Proposition 4.1 (see, for example, [15])that Σ k ⊃ Σ k +1 for all k ≥
0, and (cid:101) B ∞ = (cid:84) k ≥ Σ k . It follows that Σ k −−−−→ k →∞ (cid:101) B ∞ inHausdorff metric. It is therefore sufficient to prove thatlim sup k →∞ dist H ( (cid:101) σ N + k , Σ k ) = 0 . (36)Recall that f | S is a factor of the toral automorphism A : T → T defined in(17), and the factor map F is given in (18). The Markov partition for A on T is given in Figure 7 (see [12] and [18] for more details). This Markov partition iscarried to a Markov partition on S for f by the factor map F . PECTRUM OF QUANTUM ISING QUASICRYSTAL 29 (0,0) (0, 1/2) (1,0)(0,1) (1,1) • ••• •
Figure 7.
The Markov partition for T | S (picture taken from [18]).Let Λ be the line segment along γ that connects the singularities P and P . Then Λ is precisely the set of those points on γ whose forward orbit underiterations of f | S is bounded. The set F − (Λ ) is the line segment connecting(0 ,
0) and (0 , /
2) in the Markov partition shown in Figure 7. This set is denselyintersected by the stable manifold on T of the point (0 , F to a dense subset of Λ formed by intersections of Λ with thestrong-stable manifold on S of the point P .Let ϑ be the curve of period-two periodic points for f , passing through P , asdefined in (19). Recall that W s1 denotes the stable manifold to ϑ . Since ϑ has twosmooth branches connecting at P , W s1 can be realized as two smooth manifolds,call them W s1 ,j , j = 1 ,
2, that connect smoothly along the strong-stable manifold of P on S . Lemma 5.18.
Let Λ r be a compact line segment along γ r which contains the in-tersection of γ r with the center-stable manifolds. Assume also that the endpoints of Λ r belong to this intersection. Then for all r ≈ but not equal to one, there existsa set (cid:8) G ir (cid:9) i ∈ N of open, mutually disjoint subintervals of Λ r (we call them gaps),such that (cid:101) B ∞ ⊂ Λ r \ (cid:83) i G ir , and the collection of endpoints of all G ir is a densesubset of (cid:101) B ∞ . Moreover, for each i , one of the endpoints of G ir belongs to W s1 , ,and the other to W s1 , .Proof. Let r ∈ (0 , r ∈ (1 − r , r ), r (cid:54) = 1, Λ r intersectsthe center-stable manifolds transversely. Since Λ r =1 intersects the strong-stablemanifold of P transversely (in a dense set of points), as soon as r is slightlyperturbed, a gap opens with one endpoint in W s1 , , the other in W s1 , (see Figure8). This gap persists for all r ∈ (1 − r , r ) (i.e. as long as Λ r intersects thecenter-stable manifolds transversely).In order to show that the endpoints of these gaps form a dense subset of (cid:101) B ∞ , itis enough to show that no point inside of a gap belongs to (cid:101) B ∞ . This follows from, S Λ Λ r W s1 , W s1 , strong-stable manifoldgap Figure 8. h ( x ) h ( x ) x x W u ( x i ) τ W s (( h ( x )) W s ( h ( x )) h Figure 9. for example, [11, Theorem 5.22] (in fact, the strong-stable and strong-unstablemanifolds of the eight points that are born from the singularities { P , . . . , P } formboundaries of the Markov partition on S V , V > (cid:3)
Fix (cid:15) > C , . . . , C m be an open cover of (cid:101) B ∞ , with diam ( C i ) ≤ (cid:15) , i = 1 , . . . , m . It follows that for all k sufficiently large, Σ k is entirely contained in (cid:83) i C i .Now, for any j ∈ { , . . . , m } , pick a gap whose endpoints lie inside of C j . Callone endpoint e , and the other e . Assume that e lies on S V , and e on S V (ofcourse, V , V > ϑ ∩ S V = { p , q } , and ϑ ∩ S V = { p , q } (here f ( p i ) = q i , f ( q i ) = p i ). Say p and p lie on the same one of the two branches of ϑ , and e ∈ W s ( p ). Then e ∈ W s ( q ).Observe that if x = ( x, x/ (2 x − , x ) ∈ ϑ , then f ( x ) = ( x/ (2 x − , x, x/ (2 x − x (cid:54) = P , either | π ( x ) | < | π ◦ f ( x ) | <
1. It follows that | π ( p i ) | < | π ( q i ) | <
1. Since (cid:12)(cid:12) f k ( p ) − f k ( e ) (cid:12)(cid:12) −−−−→ k →∞ (cid:12)(cid:12) f k ( q ) − f k ( e ) (cid:12)(cid:12) −−−−→ k →∞ , and | V − V | is small provided that (cid:15) is sufficiently small (hence | p − p | and | q − q | are small), it follows that, for all k sufficiently large, either (cid:12)(cid:12) π ◦ f k ( e ) (cid:12)(cid:12) < (cid:12)(cid:12) π ◦ f k ( e ) (cid:12)(cid:12) <
1. Therefore, for all k sufficiently large, (cid:101) σ N + k ∩ C j (cid:54) = ∅ , proving(36). (cid:3) Now define (cid:101) B −∞ = (cid:110) λ ∈ ( −∞ ,
0] : | λ | ∈ (cid:101) B ∞ (cid:111) and (cid:101) σ − k = { λ ∈ ( −∞ ,
0] : | λ | ∈ (cid:101) σ k } . Take B ∞ = (cid:101) B ∞ ∪ (cid:101) B −∞ . Then σ k in (11) is precisely (cid:101) σ k ∪ (cid:101) σ − k , and so σ k −−−−→ k →∞ B ∞ in Hausdorff metric.This completes the proof of (i) of Theorem 2.1. In what follows, we prove propertiesof B ∞ stated in (ii)–(iv) of Theorem 2.1.5.4. Proof of Theorem 2.1-(ii).
This is a direct consequence of Propositions4.12 and 5.1.
PECTRUM OF QUANTUM ISING QUASICRYSTAL 31
Proof of Theorem 2.1-(iii).
Let M be a smooth two-dimensional Riemann-ian manifold and Λ ⊂ M a basic set for f ∈ Diff ( M ). Let { f α } ⊂ Diff ( M )depending continuously on α ∈ R + , with f = f . Then there exists β > α ≤ β , f α has a basic set Λ α near Λ = Λ. Let { τ α } , α ∈ R + , be a family ofsmooth compact regular curves depending continuously on α in the C -topology.Assume also that τ intersects W s (Λ ) transversely. Then there exists β ∈ R + , suchthat for all α ∈ [0 , β ), τ α intersects W s (Λ α ) transversely. Hence we may define theholonomy map h α : τ α ∩ W s (Λ α ) → Λ α (37)by sliding points along the stable manifolds to an unstable one (see Figure 9). Thenlocally h loc α and its inverse are well-defined, it is a homeomorphism onto its image,and, together with its inverse, is Lipschitz (Lipschitz continuity follows easily fromSection B.1.4). Proposition 5.19.
There exists β > such that the Lipschitz constant for h loc α and its inverse can be chosen uniformly for all α ∈ [0 , β ) . To prove Proposition 5.19, one proceeds by a (rather standard) technique thatwas first introduced in [1]. The proof is sketched in Appendix C.Recall that a morphism of metric spaces H : ( M , d ) → ( M , d ) is said tobe ( C, ν )-H¨older continuous provided that for all x, y ∈ M , d ( H ( x ) , H ( y )) ≤ Cd ( x, y ) ν . We have the following, due to J. Palis and M. Viana [57, Theorem B]. Proposition 5.20.
Let f : M → M be a C diffeomorphism on a Riemannian2-manifold and Λ ⊂ M a basic set for f with (1 , splitting. Then there exists C > and for any ν ∈ (0 , there exists U ⊂
Diff ( M ) an open neighborhood of f such that for all g ∈ U and x ∈ Λ , H g | W u ( x ) ∩ Λ and its inverse are ( C, ν ) -H¨oldercontinuous. Here H g : Λ → Λ g is the topological conjugacy (see Section B.1.1). Under the hypothesis of and with the notation from Proposition 5.19, we get
Lemma 5.21.
Let β > satisfying Proposition 5.19. There exists C > and forany ν ∈ (0 , there exists β ∈ (0 , β ) , such that for any α , α ∈ [0 , β ) , the map [ h loc α ] − ◦ H α ,α ◦ h loc α : τ α ∩ W s (Λ α ) → τ α ∩ W s (Λ α ) , where H α ,α : Λ α → Λ α is the topological conjugacy, is ( C, ν ) -H¨older continuous(see Figure 10); that is, we have the following diagram, with ( h loc α i ) ± ( i = 1 , )Lipschitz, with the same Lipschitz constant, and H α ,α H¨older continuous. τ W u ( y ) τ H α ,α ( W u ( y )) h α H α ,α h − α ◦ H α ,α ◦ h α h − α where y is such that W u ( y ) is an unstable manifold at the point y containing theimage of h α .Remark . We cannot expect any higher modulus of continuity than H¨olderin general. Indeed, we cannot expect the Hausdorff dimension of the hyperbolicsets of a family of diffeomorphisms to be constant (which would be implied if x x y = h α ( x ) y = h α ( x ) H α ,α ( y ) H α ,α ( y ) [ h α ] − ( H α ,α ( y ))[ h α ] − ( H α ,α ( y )) W u ( H α , α ( y i )) W u ( y i ) τ τ W s ( y ) W s ( y ) W s ( H α ,α ( y )) W s ( H α ,α ( y )) h α [ h α ] − H α , α [ h α ] − ◦ H α , α ◦ h α Figure 10. above instead of H¨older continuity we had Lipschitz); very simiple examples can beeasily constructed. Yet more generally, holonomy maps along the so-called centerfoliations are quite bad: see J. Milnor’s exposition of Katok’s example of so-called
Fubini nightmare in [54], as well as genericity results in [65].
Proof.
Let x ∈ τ ∩ W s (Λ α ) and h loc α ( x ) ∈ W u ( y ). Since the stable and unstablemanifolds depend continuously on the point and on the diffeomorphism, if α issufficiently close to α , then τ α ∩ W s ( H α ,α ( y )) (cid:54) = ∅ . Let x ∈ W s ( H α ,α ( y )) ∩ τ α and y = H α ,α ( y ). Now h loc α : U → W u ( y ), where U is a neighborhood of x in τ α , is defined and Propositions 5.19 and 5.20 can be applied together. (cid:3) Recall from (22): γ r ( λ ) = (cid:18) λ − (1 + J )2 J , λ − (1 + r J )2 rJ , r r (cid:19) . Hence γ r lies in the plane Π r = (cid:26) z = 1 + r r (cid:27) . Let r ∈ (0 ,
1) be as in Proposition 5.1. Let r ∈ (1 − r , r ), r (cid:54) = 1. Fix x ∈ γ r whose forward orbit under f is bounded. Pick δ small, with 0 < δ < V ,and let Γ be a compact segment along γ r containing x , with endpoints lying on S V − δ and S V + δ (note, x may be an endpoint of Γ). Then Γ intersects the center-stable manifolds, as well as the surfaces S V , V ∈ [ V − δ, V + δ ], transversely. For V ∈ [ V − δ, V + δ ], let τ V denote the projection of Γ onto S V along the plane Π r .Then τ V is a smooth, compact regular curve in S V intersecting W s (Ω V ) transversely(transversality with the center-stable manifolds follows from Proposition 5.1). Let PECTRUM OF QUANTUM ISING QUASICRYSTAL 33 x x Π r Γ τ V + δ τ V τ V − δ ϑ x ϑ x C Figure 11. C = τ V ∩ W s (Ω V ). For every x ∈ C , let ϑ x = W cs ( x ) ∩ Π r ∩ (cid:91) V ∈ [ V − δ,V + δ ] S V , where W cs ( x ) is the center-stable manifold containing x . Then (see Figure 11)C1. { τ V } form a smooth foliation of Π;C2. ϑ x is a smooth compact regular curve with endpoints lying in S V − δ and S V + δ ;C3. { ϑ x } x ∈C intersects { τ V } V ∈ [ V − δ,V + δ ] and Γ uniformly transversely, and thecurves ϑ x depend continuously on x in the C -topology (see Remark 4.11);C4. For δ > { τ V } V ∈ [ V − δ,V + δ ] and { f V } V ∈ [ V − δ,V + δ ] satisfythe hypothesis of Lemma 5.21. In particular, there exists C > ν ∈ (0 ,
1) there exists 0 < (cid:15) < δ , such that the map
C (cid:51) x (cid:55)→ τ V ∩ W s (Ω V ) definedby projecting points along the curves { ϑ x } x ∈C is ( C, ν )-H¨older continuous forall V ∈ [ V − (cid:15), V + (cid:15) ].Now, from C1–C4 it follows that there exists a sufficiently small neighborhood U of x in τ V and (cid:101) C >
0, such that the map U ∩ C (cid:51) x (cid:55)→ Γ defined by projectingpoints along the curves { ϑ x } x ∈ U ∩C is ( (cid:101) C, ν )-H¨older continuous (see [79, Lemma 3.5]for technical details). Hence dim locH (Γ , x ) = dim locH ( C , x ) . (38) On the other hand, dim locH ( C , x ) = dim H ( W u loc ( h ( x ))) = h u (Ω V ) , where h : C → Ω V is a holonomy map as defined in (37), and h u is defined in SectionB.1.5 as the Hausdorff dimension of Ω V along leafs of the unstable lamination W u (Ω V ). Now, h u (Ω V ) depends continuously (in fact analytically, as we shall seebelow) on V . It follows that the local Hausdorff dimension is continuous over B ∞ . Proposition 5.23 ([11, Theorem 5.23]) . Let γ be an analytic curve in (cid:83) V > S V parameterized on (0 , . Then h s,u (cid:0) Ω γ ( t ) (cid:1) : (0 , → R is analytic with valuesstrictly between zero and one. Proposition 5.24 ([19, Theorem 1]) . The Hausdorff dimension of Ω V is right-continuous at V = 0 : lim V → + h u (Ω V ) = 1 . Now, for r ∈ (1 − r , r ), letΓ r = (cid:110) λ : O + f ( γ r ( λ )) is bounded (cid:111) . As we have already seen, Γ r is a Cantor set; in particular it contains limit points.On the other hand, γ r is an analytic curve that intersects S V for every V ∈ [0 , ∞ ).Combining this with (38), Propositions 5.23 and 5.24, we get that the local Haus-dorff dimension is non-constant over the spectrum.Also, (38) in combination with Proposition 5.23 shows that the local Hausdorffdimension at every point of B ∞ is strictly between zero and one, hence so is theglobal Hausdorff dimension. It follows that the Lebesgue measure of B ∞ is zero.5.6. Proof of Theorem 2.1-iv.
Observe that the line γ ( J ,J ) is continuous in theparameters ( J , J ). From constructions carried out in the previous section, it isevident that dim H ( B ∞ ( J , J )) is also continuous in the parameters ( J , J ). Also,by Proposition 5.24, continuity extends to the pure case J = J . Acknowledgement
I wish to express gratitude to my dissertation advisor, Anton Gorodetski, whooriginally introduced me to this problem and whose guidance and support havebeen essential for completion of this project.I also wish to thank Michael Baake, Jean Bellissard, Alexander Chernyshev,David Damanik, Uwe Grimm, Svetlana Jitomirskaya, Ron Lifshitz and LaurentRaymond for meaningful and illuminating discussions.A special thanks to Michael Baake, David Damanik and Anton Gorodetski fortaking the time to read and comment on a draft.Finally, I wish to thank the anonymous referees for their very helpful suggestionsand remarks.
Appendices
None of what is presented in the following appendices is new. In particular, allof appendix B is by now part of the classical theory of hyperbolic and partiallyhyperbolic dynamical systems (and references are given to comprehensive surveys). Appendix A. Normal hyperbolicity of six-cycles throughsingularities of S and symmetries of f A.1.
Normal hyperbolicity of six-cycles through singularities of S . Ashas been mentioned above, S contains four conic singularities; explicitly, they are P = (1 , , , P = ( − , − , , P = (1 , − , − , P = ( − , , − . (39)The point P is fixed under f , while P , P and P form a three cycle: P f (cid:55)−→ P ; P f (cid:55)−→ P f (cid:55)−→ P f (cid:55)−→ P (which can be verified via direct computation).For each i ∈ { , . . . , } , there is a smooth curve ρ i which does not contain anyself-intersections, passing through the singularity P i , such that ρ i \ P i is a disjointunion of two smooth curves—call them ρ li and ρ ri —with the following properties: • ρ l,ri ⊂ (cid:83) V > S R V ; • f ( ρ l ) = ρ r and f ( ρ r ) = f ( ρ l ). In particular, points of ρ l,r are periodic ofperiod two, and ρ is fixed under f ; • The six curves ρ l,ri , i = 2 , ,
4, form a six cycle under f . In particular,points of ρ l,ri , i = 2 , ,
4, are periodic of period six, and hence for i = 2 , , ρ i is fixed under f .The curve ρ is given explicitly by ρ = (cid:26)(cid:18) x, x x − , x (cid:19) : x ∈ ( −∞ , / ∪ (1 / , ∞ ) (cid:27) . (40)Expressions for the other three curves can be obtained from (40) using symmetriesof f to be discussed below.It follows via a simple computation that for any i ∈ { , . . . , } and any point p ∈ ρ l,ri , the eigenvalue spectrum of Df p is { , λ ( p ) , /λ ( p ) } with 0 < | λ ( p ) | <
1, where Df p denotes the differential of f at the point p . The eigenspace corresponding tothe eigenvalue 1 is tangent to ρ i at p . At P i , the eigenvalue spectrum of Df P i is ofthe same form, and as above, the eigenspace corresponding to the unit eigenvalue istangent to ρ i at P i . It follows that the curves { ρ i } i ∈{ ,..., } are normally hyperbolicone-dimensional submanifolds of R , as defined in Section B.3.For V > S R V (cid:84) ( ρ ∪ · · · ∪ ρ ) consists of eight hyperbolic periodic points for f which are fixed hyperbolic points for f . The stable manifolds to these points forma dense sublamination of the stable lamination W s (Ω V ) = (cid:91) x ∈ Ω V W s ( x ) , the union of global stable manifolds to points in Ω V , the nonwandering set for f on S V from Theorem 4.6. Moreover, this sublamination forms the boundary of thestable lamination. For details, see [11, 12, 18]. This extends to the center-stablelamination: the stable manifolds to the normally hyperbolic curves ρ , . . . , ρ forma dense sublamination of the (two-dimensional) center-stable lamination and formsthe boundary of this lamination. In particular, if γ is a smooth curve intersectinga center-stable manifold, and this intersection is not isolated, then in an arbitrarilysmall neighborhood of this intersection, γ intersects the stable manifolds of all eightcurves: (cid:110) ρ l,ri (cid:111) i ∈{ ,..., } . A.2.
Symmetries of f . The following discussion is taken from [21]; however, whatfollows does not appear in [21] as new results but a recollection of what is known.In particular, the reader should consult [4, 63] and references therein, as well asearlier (and original) works [45–48].Let us denote the group of symmetries of f by G sym , and the group of reversingsymmetries of f by G rev ; that is, G sym = (cid:8) s ∈ Diff( R ) : s ◦ f ◦ s − = f (cid:9) , (41)and G rev = (cid:8) s ∈ Diff( R ) : s ◦ f ◦ s − = f − (cid:9) , (42)where Diff( R ) denotes the set of diffeomorphisms on R .Observe that G rev (cid:54) = ∅ . Indeed, s ( x, y, z ) = ( z, y, x )(43)is a reversing symmetry of f , and hence also of f . Hence f is smoothly conjugateto f − . It follows (see Appendix A.1) that forward-time dynamical properties of f , as well as the geometry of dynamical invariants (such as stable manifolds) aremapped smoothly and rigidly to those of f − . That is, forward-time dynamics of f is essentially the same as its backward-time dynamics.The group G sym is also nonempty, and more importantly, it contains the followingdiffeomorphisms: s : ( x, y, z ) (cid:55)→ ( − x, − y, z ) ,s : ( x, y, z ) (cid:55)→ ( x, − y, − z ) , (44) s : ( x, y, z ) (cid:55)→ ( − x, y, − z ) . Notice that s i are rigid transformations. Also notice that s i ( P ) = P i , and since s i is in fact a smooth conjugacy, we must have s i ( ρ ) = ρ i . (45)For a more general and extensive discussion of symmetries and reversing symmetriesof trace maps, see [4]. Appendix B. Background on uniform, partial and normalhyperbolicity
B.1.
Properties of locally maximal hyperbolic sets.
A more detailed discus-sion can be found in [30, 31, 34–36].A closed invariant set Λ ⊂ M of a diffeomorphism f : M → M of a smoothmanifold M is called hyperbolic if for each x ∈ Λ, there exists the splitting T x Λ = E sx ⊕ E ux invariant under the differential Df , and Df exponentially contracts vectorsin E sx and exponentially expands vectors in E ux . The set Λ is called locally maximal if there exists a neighborhood U of Λ such thatΛ = (cid:92) n ∈ Z f n ( U ) . (46)The set Λ is called transitive if it contains a dense orbit. It isn’t hard to prove thatthe splitting E sx ⊕ E ux depends continuously on x ∈ Λ, hence dim( E s,ux ) is locallyconstant. If Λ is transitive, then dim( E s,ux ) is constant on Λ. We call the splitting E sx ⊕ E ux a ( k sx , k ux ) splitting if dim( E s,ux ) = k s,u , respectively. In case Λ is transitive,we shall simply write ( k s , k u ). Definition B.1.
We call Λ ⊂ M a basic set for f ∈ Diff r ( M ), r ≥
1, if Λ is alocally maximal invariant transitive hyperbolic set for f .Suppose Λ is a basic set for f with (1 ,
1) splitting. Then the following holds.B.1.1.
Stability.
Let U be as in (46). Then there exists U ⊂
Diff ( M ) open, con-taining f , such that for all g ∈ U ,Λ g = (cid:92) n ∈ Z g n ( U )(47)is g -invariant transitive hyperbolic set; moreover, there exists a (unique) homeo-morphism H g : Λ → Λ g such that H g ◦ f | Λ = g | Λ g ◦ H g ;(48)that is, the following diagram commutes.Λ ΛΛ g Λ g f H g H g g Also H g can be taken arbitrarily close to the identity by taking U sufficientlysmall. In this case g is said to be conjugate to f , and H g is said to be the conjugacy .B.1.2. Stable and unstable invariant manifolds.
Let (cid:15) > x ∈ Λdefine the local stable and local unstable manifolds at x : W s(cid:15) ( x ) = { y ∈ M : d ( f n ( x ) , f n ( y )) ≤ (cid:15) for all n ≥ } ,W u(cid:15) ( x ) = { y ∈ M : d ( f n ( x ) , f n ( y )) ≤ (cid:15) for all n ≤ } . We sometimes do not specify (cid:15) and write W s loc ( x ) and W u loc ( x )for W s(cid:15) ( x ) and W u(cid:15) ( x ), respectively, for (unspecified) small enough (cid:15) >
0. For all x ∈ Λ, W s,u loc ( x ) is an embedded C r disc with T x W s,u loc ( x ) = E s,ux . The global stable and global unstable manifolds W s ( x ) = (cid:91) n ∈ N f − n ( W s loc ( x )) and W u ( x ) = (cid:91) n ∈ N f n ( W u loc ( x ))(49)are injectively immersed C r submanifolds of M . Define also the stable and unstablesets of Λ: W s (Λ) = (cid:91) x ∈ Λ W s ( x ) and W u (Λ) = (cid:91) x ∈ Λ W u ( x ) . (50)If Λ is compact, there exists (cid:15) > x, y ∈ Λ, W s(cid:15) ( x ) ∩ W u(cid:15) ( y )consists of at most one point, and there exists δ > d ( x, y ) < δ , x, y ∈ Λ, then W s(cid:15) ( x ) ∩ W u(cid:15) ( y ) (cid:54) = ∅ . If in addition Λ is locally maximal, then W s(cid:15) ( x ) ∩ W u(cid:15) ( y ) ∈ Λ. The stable and unstable manifolds W s,u loc ( x ) depend continuously on x in the sensethat there exists Φ s,u : Λ → Emb r ( R , M ) continuous, with Φ s,u ( x ) a neighborhoodof x along W s,u loc ( x ), where Emb r ( R , M ) is the set of C r embeddings of R into M [35, Theorem 3.2].The manifolds also depend continuously on the diffeomorphism in the followingsense. For all g ∈ Diff r ( M ) C r close to f , define Φ s,ug : Λ g → Emb r ( R , M ) as wedefined Φ s,u above. Then define˜Φ s,ug : Λ → Emb r ( R , M )by ˜Φ s,ug = Φ s,ug ◦ H g . Then ˜Φ s,ug depends continuously on g [35, Theorem 7.4].B.1.3. Fundamental domains.
Along every stable and unstable manifold, one canconstruct the so-called fundamental domains as follows. Let W s ( x ) be the stablemanifold at x . Let y ∈ W s ( x ). We call the arc γ along W s ( x ) with endpoints y and f − ( y ) a fundamental domain . The following holds. • f ( γ ) ∩ W s ( x ) = y and f − ( γ ) ∩ W s ( x ) = f − ( y ), and for any k ∈ Z , if k < −
1, then f k ( γ ) ∩ W s ( x ) = ∅ ; if k > f k ( γ ) ∩ W s ( x ) = ∅ iff x (cid:54) = y ; • For any z ∈ W s ( x ), if for some k ∈ N , f k ( z ) lies on the arc along W s ( x )that connects x and y , then there exists n ∈ N , n ≤ k , such that f n ( z ) ∈ γ .Similar results hold for the unstable manifolds.B.1.4. Invariant foliations.
A stable foliation for Λ is a foliation F s of a neighbor-hood of Λ such that(1) for each x ∈ Λ, F ( x ), the leaf containing x , is tangent to E sx ;(2) for each x sufficiently close to Λ, f ( F s ( x )) ⊂ F s ( f ( x )).An unstable foliation F u is defined similarly.For a locally maximal hyperbolic set Λ ⊂ M for f ∈ Diff ( M ), dim( M ) = 2,stable and unstable C foliations with C leaves can be constructed; in case f ∈ Diff ( M ), C invariant foliations exist (see [56, Section A.1] and the referencestherein).B.1.5. Local Hausdorff and box-counting dimensions.
For x ∈ Λ and (cid:15) >
0, considerthe set W s,u(cid:15) ∩ Λ. Its Hausdorff dimension is independent of x ∈ Λ and (cid:15) > h s,u (Λ) = dim H ( W s,u(cid:15) ( x ) ∩ Λ) . (51)For properly chosen (cid:15) >
0, the sets W s,u(cid:15) ( x ) ∩ Λ are dynamically defined Cantorsets, so h s,u (Λ) < h s,u depends continuously on the diffeomorphismin the C -topology [53]. In fact, when dim( M ) = 2, these are C r − functions of f ∈ Diff r ( M ), for r ≥ Box (Γ). Thendim H ( W s,u(cid:15) ( x ) ∩ Λ) = dim
Box ( W s,u(cid:15) ( x ) ∩ Λ)(see [53, 72]). B.2.
Partial hyperbolicity.
For a more detailed discussion, see [32, 58].An invariant set Λ ⊂ M of a diffeomorphism f ∈ Diff r ( M ), r ≥
1, is called partially hyperbolic (in the narrow sense) if for each x ∈ Λ there exists a splitting T x M = E sx ⊕ E cx ⊕ E ux invariant under Df , and Df exponentially contracts vectorsin E sx , exponentially expands vectors in E ux , and Df may contract or expand vectorsfrom E cx , but not as fast as in E s,ux . We call the splitting ( k sx , k cx , k ux ) splitting ifdim( E s,c,ux ) = k s,c,ux , respectively. We shall write ( k s , k c , k u ) if the dimension ofsubspaces does not depend on the point.B.3. Normal hyperbolicity.
For a more detailed discussion and proofs see [36]and also [58].Let M be a smooth Riemannian manifold, compact, connected and withoutboundary. Let f ∈ Diff r ( M ), r ≥
1. Let N be a compact smooth submanifoldof M , invariant under f . We call f normally hyperbolic on N if f is partiallyhyperbolic on N . That is, for each x ∈ N , T x M = E sx ⊕ E cx ⊕ E ux with E cx = T x N . Here E s,c,ux is as in Section B.2. Hence for each x ∈ N one canconstruct local stable and unstable manifolds W s(cid:15) ( x ) and W u(cid:15) ( x ), respectively, suchthat(1) x ∈ W s loc ( x ) ∩ W u loc ( x );(2) T x W s loc ( x ) = E s ( x ), T x W u loc ( x ) = E u ( x );(3) for n ≥ d ( f n ( x ) , f n ( y )) −−−−→ n →∞ y ∈ W s loc ( x ) ,d ( f − n ( x ) , f − n ( y )) −−−−→ n →∞ y ∈ W u loc ( x ) . (For the proof see [58, Theorem 4.3]). These can then be extended globally by W s ( x ) = (cid:91) n ∈ N f − n ( W s loc ( x ));(52) W u ( x ) = (cid:91) n ∈ N f n ( W u loc ( x )) . (53)The manifold W s ( x ) is referred to as the strong-stable manifold , while W u ( x ) iscalled the strong-unstable manifold; sometimes to emphasize the point x , we add at x .Set W cs loc ( N ) = (cid:91) x ∈ N W s loc ( x ) and W cu loc ( N ) = (cid:91) x ∈ N W u loc ( x ) . (54) Theorem B.2 (Hirsch, Pugh and Shub [36]) . The sets W cs loc ( N ) and W cu loc ( N ) ,restricted to a neighborhood of N , are smooth submanifolds of M . Moreover,(1) W csloc ( N ) is f -invariant and W culoc is f − -invariant;(2) N = W csloc ( N ) (cid:84) W culoc ( N ) ;(3) For every x ∈ N , T x W cs , culoc ( N ) = E s,ux ⊕ T x N ;(4) W csloc ( N ) ( W culoc ( N ) ) is the only f -invariant ( f − -invariant) set in a neigh-borhood of N ; (5) W csloc ( N ) (respectively, W culoc ( N ) ) consists precisely of those points y ∈ M such that for all n ≥ (respectively, n ≤ ), d ( f n ( x ) , f n ( y )) < (cid:15) for some (cid:15) > .(6) W cs , culoc ( N ) is foliated by { W s , uloc ( x ) } x ∈ N . Appendix C. Background results
We prove here some background results that follow from rather general principlesin dynamical systems.C.1.
Proof of Proposition 5.19: sketch of main ideas.
To prove Proposition5.19, one proceeds by a (rather standard) technique that was first introduced in [1].Let us sketch the proof below.First suppose τ α = τ = τ for all α . Let x ∈ τ ∩ W s (Λ) and γ an open arc along τ containing x such that h loc and its inverse are defined along γ . Let U be anopen neighborhood of Λ such that Λ is maximal in U , and U can be foliated intostable and unstable foliations. There exists k ∈ N such that for all α , f k α ( x ) ∈ U .Assuming γ is sufficiently short, we also have f k α ( γ ) ⊂ U .To simplify notation, let us write f for f . Let γ s = f k ( γ ) and let γ u be theunstable manifold such that f k ◦ h ( x ) ∈ γ u . Let ˜ h : γ s → γ u be the inducedholonomy map: ˜ h ( x ) = f k ◦ h loc ◦ f − k . By the C stable foliation, ˜ h may be considered as the restriction of a C map F : γ s → γ u to the set f k ( γ ∩ W s (Λ)). Then for all x, y ∈ γ s sufficiently close,there exists k = k ( x, y ) ∈ N such that the arc along f k ( γ s ) (respectively, f k ( γ u ))connecting the points f k ( x ) , f k ( y ) (respectively, f k ( F ( x )) , f k ( F ( y ))) belongs to U ,and (cid:34) dist f k ( γ s ) (cid:0) f k ( x ) , f k ( y ) (cid:1) dist f k ( γ u ) ( f k ( F ( x )) , f k ( F ( y ))) (cid:35) ± ≤ , (55)where dist β ( a, b ) denotes the distance between points a and b along the curve β (the number 2 is not significant; anything larger than 1 will work). Hence it isenough to provide an estimate, independent of k , for (cid:20) dist γ s ( x, y ) dist f k ( γ s ) ( f k ( x ) , f k ( y )) (cid:21) ± (cid:34) dist f k ( γ u ) (cid:0) f k ( F ( x )) , f k ( F ( y )) (cid:1) dist γ u ( F ( x ) , F ( y )) (cid:35) ± . (56)In order to estimate (56), it is enough to estimate (cid:13)(cid:13) Df k | γ s ( x,y ) (cid:13)(cid:13)(cid:13)(cid:13) Df k | γ u ( F ( x ) ,F ( y )) (cid:13)(cid:13) , where γ s,u ( a, b ) is the arc along γ s,u with endpoints a, b . After taking log, oneestimates the latter by estimating k (cid:88) j =0 (cid:12)(cid:12)(cid:13)(cid:13) Df j | γ s ( x,y ) (cid:13)(cid:13) − (cid:13)(cid:13) Df j | γ u ( F ( x ) ,F ( y )) (cid:13)(cid:13)(cid:12)(cid:12) . The sum above is majorized by a geometric series, and hence admits an upperbound L for all k . One shows that the bound in (55) and the bound L , for L sufficiently large, hold for all f α , α ∈ (0 , β ), with β sufficiently small (this followsfrom continuous dependence of f α and Df α on α ).Finally, small C perturbations of τ do not destroy these bounds. References [1] D. V. Anosov and Ya. G. Sinai. Some Smooth Ergodic Systems.
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