aa r X i v : . [ m a t h . SP ] F e b ENTROPY OF LOGARITHMIC MODES
SURESH ESWARATHASAN
Abstract.
Consider (
M, g ) a compact, boundaryless Riemannian manifold admitting anAnosov geodesic flow. Let ε < min { , λ max } where λ max is the maximal expansion rate for( M, g ). We study the semiclassical measures µ sc of ε -logarithmic modes, which are quasi-modes spectrally supported in intervals of width ε ~ | log ~ | , of the Laplace-Beltrami operatoron M . Under a technical assumption on the support of a signed measure related to µ sc ,we show that the lower bound for the Kolmogorov-Sinai entropy of µ sc generalizes that ofAnanthamaran-Koch-Nonnenmacher [2]. A feature of our results is the quantitative natureof our bounds on hyperbolic manifolds.Via previous results of the author with Nonnenmacher [18] and Silberman [19], we provideanother demonstration of the existence of ε -logarithmic modes whose semiclassical measurespossess zero entropy ergodic components yet have other components with positive entropy. Introduction
Motivations.
Equidistribution and Eigenfunctions.
The semiclassical correspondence principle inquantum mechanics states that in the high-energy limit, the long-time classical mechanicsshould be reflected in the corresponding quantum phenomena. In this article, we focuson the case when the classical system is said to be chaotic. In particular, we focus onthe geometric setting of (
M, g ) being a compact, boundaryless, Riemannian manifold thatexhibits an Anosov geodesic flow g t (a slight abuse of notation) on the unit sphere bundle;compact manifolds with negative sectional curvatures form a rich class of examples. Notethat the dynamical system generated by Anosov flows g t can be thought of as the “strongest”form of chaos, at least in the mathematical sense. The considered quantum Hamiltonian isthe semiclassical Laplace-Beltrami operator − ~ ∆ g where ~ is proportional to the Planckconstant.The Quantum Unique Ergodicity (QUE) conjecture of Rudnick-Sarnak [31] states thateigenfunctions of − h ∆ g , say functions ψ ~ satisfying − ~ ∆ g ψ ~ = Eψ ~ for E > M with negative sectional curvatures become equidistributed in this high-energylimit. More specifically, for L ( M, d vol g )-normalized eigenfunctions { ψ ~ } ~ , the probabilitydensities | ψ ~ | d vol g approach (in a weak-* sense) the Riemannian volume measure d vol g uniquely as ~ →
0. In fact, a stronger property should hold for their Wigner distributions W ~ , a function on the cotangent bundle T ∗ M which simultaneously describes the localizationof the sequence { ψ ~ } ~ in position and momentum.The Wigner distribution associated to a L ( M, d vol g )-normalized sequence of states { Ψ ~ } ~ is defined as W ~ ( a ) := h Op ~ ( a )Ψ ~ , Ψ ~ i (1.1.1) Date : February 24, 2021. where a ∈ C ∞ ( T ∗ M ) and Op ~ ( • ) is a quantization procedure that associates any smoothphase-space function a to a bounded operator on L ( M, d vol g ); this procedure is describedin further detail in Section 2. We call semiclassical measures µ sc the limit points of thesequence { W ~ } ~ in the weak-* topology on the space of distributions. It is well-known[40] that semiclassical measures µ sc associated to eigenfunctions are g t -invariant and aresupported on the energy shell E := S ∗ E M . QUE in its fullest strength states the following:in the case of eigenfunctions on manifolds with negative sectional curvatures, W ~ → ∗ dL with dL being the Liouville measure on E := { ( x, ξ ) ∈ T ∗ M | k ξ k g ( x ) = E } where k • k g is the norm on T ∗ x M generated by the metric g . This more properly quantifies the word“equidistribution”. An important measure of localization, and is central to this manuscript,is the notion of Kolmogorov-Sinai entropy to which we dedicate a lengthier discussion inSection 1.2.As of the writing of this article, only a few cases of QUE have been successfully resolved:the so-called Arithmetic Quantum Unique Ergodicity proven by Lindenstrauss [28] in thecompact case and Soundararajan [36] for non-compact. These results take place on congru-ence surfaces which carry additional number-theoretic structure and admit operators whichenjoy these symmetries, namely Hecke operators; see also the work of Silberman-Venkateshfor locally symmetric spaces [34]. Specifically, Arithmetic QUE holds for joint eigenfunctionsof the Laplace-Beltrami operator and the Hecke operators. For the model of the quantumcat map on the 2-torus and again “arithmetic” joint eigenfunctions, this was established byKurlberg-Rudnick [26]. In the context of random bases on spheres, see the work of Zelditch[38].The looser notion of quantum ergodicity , as established by Schnirelman-Zelditch-Colinde Verdi´ere [33, 37, 13], states the existence of a density-one subsequence of { ψ ~ } ~ thatequidistributes and holds in greater generality: on compact manifolds exhibiting ergodic(with respect dL ) geodesic flows. Counterexamples, albeit appearing in inequivalent modelssuch as the Bunimovich stadium [24] and the quantum cat map [20, 21], have utilized spectralmultiplicities (or degeneracies, as they will be referred to in upcoming discussion) in effectiveways to demonstrate the existence of exceptional subsequences and therefore the failure ofQUE. Moreover, Arithmetic QUE was disproved by Kelmer [25] on higher dimensional torialbeit without the use large multiplicities. Note that quantum ergodicity was established onbilliards with ergodic flows in [22, 39] and cat maps in [10].In the context of quantum ergodicity (unique or not) on negatively curved surfaces, itis important to emphasize two relatively recent results that were tremendous steps in ourunderstanding of high-frequency spectral asymptotics. The first was the breakthrough ofDyatlov-Jin [16], underpinned by an earlier breakthrough of Bourgain-Dyatlov [6] on res-onance gaps for hyperbolic surfaces with funnels, proving that sequences of eigenfunctionson compact hyperbolic surfaces must delocalize across all of position space in the followingsense: Supp µ sc = E . This was later generalized to surfaces of negative sectional curvaturesby Dyatlov-Jin-Nonnenmacher [17].1.1.2. Spectral degeneracies and quasimodes.
The Laplace-Beltrami operator on manifoldsexhibiting Anosov geodesic flows is not expected to have large spectral degeneracies, there-fore leaning us towards the equidistribution phenomena. In response, one can introduce arelated form of this degeneracy by considering “large” linear combinations of eigenfunctions, -LOGARITHMIC MODES 3 functions that act as approximations to true eigenfunctions and carry the moniker of quasi-modes ; the order of approximation can be quantified with the notion of width and is givenin the following definition:
Definition (Quasimodes) . For a given energy level
E >
0, we say that a semiclassicalfamily { Ψ ~ } ~ of L -normalized states is a family of quasimodes of − ~ ∆ g of central energy E and width w ( ~ ), if and only if there exists ~ ( E ) > k ( − ~ ∆ g − E )Ψ ~ k L ( M ) ≤ w ( ~ ) , ∀ ~ ≤ ~ . (1.1.3)A slightly broader definition consists in allowing the central energy to depend on ~ as well,namely considering a sequence { E ~ } ~ with E ~ → E as ~ →
0, and requiring k ( − ~ ∆ g − E ~ )Ψ ~ k L ( M ) ≤ w ( ~ ) , ∀ ~ ≤ ~ . (1.1.4)One way to construct quasimodes of central energy E ~ and width f ( ~ ) is to take linearcombinations of eigenfunctions of − ~ ∆ g with eigenvalues in the interval [ E ~ − w ( ~ ) , E ~ + w ( ~ )]. The purpose of allowing varying central energies E ~ , and which keep a certain distancefrom E , is the existence of quasimodes that partial localize and possess such energies; seeCorollary 1.3.1 and the surrounding discussion.It has proven fruitful to study the properties of quasimodes, especially its effects on equidis-tribution and QUE, demonstrating the overall role of these degeneracies in the mathematicaltheory surrounding (but certainly not limited to) the semiclassical correspondence princi-ple. Make note that quasimodes of width o ( ~ ) having semiclassical measures µ sc are also g t -invariant and supported on the energy shell E .Inspired by the work Faure-Nonnenmacher-de Bi`evre [20] on exceptional sequences ofeigenfunctions for the quantum cat map, a trio of results due to Brooks [8], Eswarathasan-Nonnenmacher [18], and Eswarathasan-Silberman [19] rigorize the passage from artificialdegeneracies to non-equidistribution in the presence of some form of classical chaos. In par-ticular on a compact surface with a geodesic that generates a closed hyperbolic geodesic,[18] constructs a sequence of quasimodes of width C h | log ~ | with central energy E and whosesemiclassical measure is a delta function on that geodesic; in fact, this result continues tohold for more general quantum Hamiltonians whose associated classical dynamics generatesat least one closed hyperbolic orbit. The works [8, 19] are particular to constant negativecurvature with the work of Eswarathasan-Silberman holding in higher dimensions. We em-phasize that the results of Dyatlov-Jin-Nonnenmacher [16, 17] on the full support of µ sc holdfor quasimodes of width o (cid:16) h | log ~ | (cid:17) . We are now in a position to introduce our main objectsof study: Definition ( ε -logarithmic modes) . For a given energy level
E > ε >
0, wesay that a semiclassical family { Ψ ~ } ~ of L normalized states is a family of ε -logarithmicmodes if there exist a uniform constant C energy >
0, central energies { E ~ } ~ with | E ~ − E | ≤ C energy ~ | log ~ | , intervals I ( ε, ~ ) := (cid:20) E ~ − ε ~ | log ~ | , E ~ + ε ~ | log ~ | (cid:21) and an ~ ( E, ε ) > ~ ≤ ~ , the modes Ψ ~ are spectrally supported in I ( ε, ~ ): that is, Π I ( ~ ) ∁ Ψ ~ = 0. In other words, { Ψ ~ } ~ is a L -normalized family whose SURESH ESWARATHASAN spectral support is contained in I ( ~ ) therefore making it a family of quasimodes for − ~ ∆ g of central energies { E ~ } ~ with width w ( ~ ) = ε ~ | log ~ | .We emphasize that a logarithmic family of modes carries three quantities of importance atany given point of reference: a spectral-type width factor ε that acts as a degree of freedom informing our linear combinations , a family of central energies { E ~ } ~ , and an energy distancefactor C energy ; as our operator of interest is the semiclassical Laplacian, we can set E = 1throughout the remainder of our manuscript. While it is tedious to mention all of thesein every summoning of the term “ ε -logarithmic modes” and hence we choose to not, eachquantity does play a non-trivial role leading to the main result: ε in the final entropy bound,energies E ~ for the partial localizing examples, and C energy for strong microlocalization whichis crucial in applying the entropic uncertainty principle. Therefore, we urge the reader tokeep these in mind.We end by noting that for such an ε -logarithmic family, these are also quasimodes of width ε m ~ m | log ~ | m for the operator ( P − E ~ ) m thanks to the spectral support property; this fact willbe used in later sections. The central energies keeping a distance of O (cid:16) ~ | log ~ | (cid:17) from E issimply for microlocalization purposes and can be loosened in the context of our main results.1.2. Main results.
We remind ourselves that in this manuscript, (
M, g ) is a d -dimensionalcompact boundaryless manifold whose geodesic flow g t satisfies the Anosov property on E = S ∗ E M = { ( x, ξ ) ∈ T ∗ M | k ξ k g ( x ) = E } . Moreover, recall that in such geometries, E admits a foliation into unstable manifolds of the flow [7, Chapter 5]. To understand better thevarious semiclassical measures µ sc of ε -logarithmic modes, we aim compute their Kolmogorov-Sinai entropies . The Kolmogorov-Sinai entropy of a g t -invariant probability measure µ is anon-negative number H KS ( µ ) that describes the long-time, average information carried by µ -typical geodesics. A more precise definition of H KS is provided in Section 4 with furtherbackground found in [7, Chapter 9].A noteworthy, yet classical, upperbound on H KS for an invariant probability measure µ is to due to Ruelle [32]: H KS ( µ ) ≤ Z E log J u ( ρ ) dµ ( ρ ) . Here, J u ( ρ ) = J u ( ρ ) := | det D ( g ) ↾ E u ( ρ ) | is the unstable Jacobian for g t at ρ ∈ E with E u ( ρ ) being the unstable manifold at ρ ∈ E ; thisquantity is related to the volume induced by the Riemannian metric g on E . The upper boundis obtained if and only if µ = dL in the case of Anosov systems [27]. When the sectionalcurvatures K ( M ) = −
1, the the measure of maximal entropy is dL and thus H KS ( µ ) ≤ d − g t -invariant probability measure µ . Thus, QUE can be reformulated as claiming H KS ( µ sc ) = d −
1, or that µ sc is the measure of maximal entropy, when K ( M ) = − λ max := lim t →∞ sup ρ ∈E t − log | det D ( g t ) ↾ E u ( ρ ) | to be the maximal expansion rate on E . The analogous definition for the maximal expansionrate on different energy shells, or slabs, is immediate. Before stating our main results, we -LOGARITHMIC MODES 5 make the reader aware of an important technical assumption described in Section 5.3 and islabeled Assumption A . Theorem
Let ( M, g ) be a d -dimensional compact manifold admitting an Anosovgeodesic flow. Let ε < min { , λ max / } . Consider an ε -logarithmic family { Ψ ~ } ~ and let µ sc be its semiclassical measure.Under Assumption A , there exists functionals µ sc and H ( µ sc ) of the semiclassical measure µ sc so that H KS ( µ sc ) ≥ Z E log J u ( ρ ) dµ sc ( ρ ) − ( d − λ max − Z E log J u ( ρ ) d µ sc ( ρ ) − H ( µ sc ) . Furthermore, µ sc and H ( µ sc ) are both monotone decreasing towards in ε . Thus in the casethat R E log J u ( ρ ) dµ sc ( ρ ) − ( d − λ max > , we have that H KS ( µ sc ) > for ε sufficiently small.For ε = o (1) in ~ , Assumption A is satisfied and we recover H KS ( µ sc ) ≥ Z E log J u ( ρ ) dµ sc ( ρ ) − ( d − λ max . The first functional µ sc is a signed measure generated by our ε -logarithmic families and isdescribed by an infinite series of matrix elements generated by ( P ( ~ ) − E ~ ) m Ψ ~ where m ∈ N .The second functional H ( µ sc ) is defined similarly to the Kolmogorov-Sinai entropy but nowwith terms µ sc ( • ) log µ sc ( • ). See Section 8.2 for the definitions of µ sc and H ( µ sc ). Thepurpose of Assumption A is to guarantee these new quantities are quantitatively controlledby ε .We now pass to a result for constant negative curvature: Corollary
Let ( M, g ) be a d -dimensional compact hyperbolic manifold and ε < min { , λ max / } . Then under Assumption A , for any family of ε -logarithmic modes { Ψ ~ } ~ we have H KS ( µ sc ) ≥ d − − e ε − d − . In particular, the Kolmogorov-Sinai entropy is positive for ε < log √ ≈ . . . . . Fur-thermore, for ε = o (1) , we recover the bound of ( d − / . We emphasize a few aspects of our results. First, the assumption of ε < min { , λ max / } comes entirely from needing to eliminate strongly scarring modes as described in Theorem1.3.1 whose semiclassical measures have entropy 0 and to guarantee that our new functionalsconverge. Second, our Theorem 1.2.1 and Corollary 1.2.2 not only generalizes [2, Theorem2.1] to logarithmic modes but makes quantitative the bound given in [1, Theorem 1.1.2]especially in the case of constant negative curvature; see the work of Rivi`ere [30] for sharperresults on eigenfunctions in two dimensions. Furthermore, we make explicit the exact widthof logarithmic modes in order to achieve positive entropy. Third, when ε = o (1), Corollary1.2.2 and [16, Theorem 1] combined state that µ sc for o (1)-logarithmic modes is both fullysupported on E and possesses entropy at least 1/2 on hyperbolic surfaces. SURESH ESWARATHASAN
It is also worth making the remark that under additional assumptions on the relationshipsbetween the semiclassical measures it may be possible to deduce an entropy bound for afamily of quasimodes { Ψ ~ } ~ with width ε ~ | log ~ | associated to central energies { E ~ } ~ . Morespecifically, consider the ε α -logarithmic family { Ψ ~ } ~ that is obtained by projecting Ψ ~ onto I ( ε α , h ) for some α <
1. Let ν sc be the semiclassical measure for { Ψ ~ } ~ and µ sc for theprojections { Ψ ~ } ~ . By Corollary 3.1.2, we know that the total variation distance between ν sc and µ sc is bounded above by 5 ε − α . A manipulation of the algebraic expressions for H KS then gives a functional comparing ν sc and µ sc . In fact, such a functional returns 0 in thecase of quasimodes of width o (cid:16) ~ | log ~ | (cid:17) , as now ν sc = µ sc and hence Theorem 1.2.1 holds forthese slightly more general quasimodes. See Section 3.1 for more on how to project modesand deduce facts on their semiclassical measures.Lastly, while Theorem 1.2.1 takes place in the context of the semiclassical Laplace-Beltramioperator − ~ ∆ g , the result should easily generalize to quantum Hamiltonians P ( ~ ) satisfyingsome basic non-degeneracy and spectral conditions near a fixed energy E > E throughout our calculations in order to resemble those for general Hamiltonians). Weleave these details to the interested reader.1.3. Partial localization.
We review some relevant results on the localization of logarith-mic quasimodes, in particular from [19]:
Theorem (Partial localization on closed, lower-dimensional, invariant subsets in con-stant negative curvature) . Let ( M, g ) be a d -dimensional compact hyperbolic manifold. Let ε > be given. Let E > be an energy level of − h ∆ , and µ F be any g t -invariant measureon E with support Γ a closed set of Hausdorff dimension strictly less than d − . Thenthere exists ε > and a family of quasimodes { Ψ ~ } h of width ε ~ | log ~ | with central energy E and with µ sc = µ Γ . Furthermore, there exists an ε -logarithmic family { Ψ ~ } h with centralenergies E ~ = E + O (cid:16) ~ | log ~ | (cid:17) where µ sc contains an atom w Γ · µ Γ in its ergodic decompositionand w Γ ≥ ε ε √ O (cid:18)(cid:16) ε ε (cid:17) (cid:19) . The implicit constant c in the expression for E ~ can be taken close from above to ε . Considering Theorem 1.3.1 and our main result Theorem 1.2.1 in tandem, we arrive atthe next corollary (and for which an analogous result holds for variable curvature surfaces `ala the main result of [18]):
Corollary (Partially localizing modes with positive entropy) . Let ( M, g ) be a d -dimensional compact hyperbolic manifold and Γ ⊂ E an invariant subset of Hausdorff di-mension strictly less than d − equipped with an invariant measure µ Γ . Let ε be as inTheorem 1.3.1.Under Assumption A , given H ∈ [0 , ( d − / , there exists ε = ε ( H , ε ) and a sequence { Ψ ~ } ~ of ε -logarithmic modes with central energies O (cid:16) ~ | log ~ | (cid:17) and whose µ sc satisfies thefollowing properties: (1) H KS ( µ sc ) ≥ H , (2) µ sc has µ Γ as an ergodic component. -LOGARITHMIC MODES 7 A closing remark regarding our results is that there is still a desire to elucidate the con-nection between degeneracies in the spectrum of chaotic geometries and QUE, which in itscurrent status is still unclear in many respects. The aim of this article was to make progressin this direction: we demonstrate that even for modes consisting of O (cid:16) ~ − ( d − | log ~ | (cid:17) eigenfunc-tions (`a la currently available Weyl asymptotics), semiclassical measures of positive entropyare attainable at least under Assumption A . At the same time, at least on hyperbolic man-ifolds, there is now conditional evidence supporting the claim that combinations generatedby intervals of width larger than log (cid:16) √ (cid:17) ~ | log ~ | return only zero entropy measures.1.4. On Assumption A.
Assumption A is in fact a statement about the support of anewly defined signed measure µ sc ; see its formula in Definition 8.2.1. In fact, we wantthat µ sc is absolutely continuous with respect to µ sc . The immediate consequences, viainvariance statements proven in Section 8.2, are advantageous and guarantee the existenceand monotonicity in ε of the new entropy-type functional H ( µ sc ). In the case of { Ψ ~ } ~ bea family of o (cid:16) ~ | log ~ | (cid:17) quasimodes, Assumption A is trivially satisfied.We emphasize that a variety of more general assumptions could have been made to ob-tain Theorem 1.2.1, say on exceptional dynamical sequences where the Shannon-McMillan-Breiman Theorem fails and which results in H ( µ sc ) being well-behaved; we chose the mostnatural and intuitive statement in the end. Certainly, one could have a final entropy bound inthe presence of no such assumptions but at the cost of little information on the ε -dependenceof our functionals µ sc and H ( µ sc ); this case, it may be possible that our lower bound is neg-ative and therefore trivial.In fact, the clever idea of Anantharaman-Nonnenmacher’s to use shift-invariance beyondthe Ehrenfest time (described in Proposition 5.2.2) to deduce information on entropies atshort times via subadditivity argument appears to hinge upon our Assumption A . In otherwords, our assumption applies to the remainder in the shift-invariance formula 5.2.2 andallows us to effectively apply the subadditivity argument after using the now well-knownEntropic Uncertainty Principle. It is not clear to us whether this indicates the limitationsof the subadditivity argument at certain ranges beyond Ehrenfest or simply a technicaloversight in our analysis.1.5. On the proof: Contributions and outline.
New contributions.
We follow the broad strokes of Anantharaman-Nonnenmacher (-Koch)’s argument [3, 2] in that we also apply the Entropic Uncertainty Principle, describedin Section 6.2, and use the sharper hyperbolic dispersive estimate stated in [2, Theorem3.5]. It is important to note that outside of our assumptions on the flow being Anosov,and on which the hyperbolic dispersive estimate (Proposition 4.3.3) crucially depends, ourmethodology is completely general . Our first biggest departure is in establishing a sharper form of the shift-invariance formulafor quantum measures (Proposition 5.2.2) associated to ε -logarithmic modes; this shouldbe contrasted with [3, Proposition 4.1] for eigenfunctions and [1, Proposition 1.3.1] for ε -logarithmic modes. Our more general invariance statements feeds into a need for more precisesubadditivity statements for entropies and pressures, themselves appearing in Sections 7 and8 and which constitute our second biggest departure . The path towards a finer description SURESH ESWARATHASAN imposes the analysis of many, yet still simple, functionals each of which are dependent onour modes { Ψ ~ } ~ and are “sub-principal” in nature. These are all laid out in Section 8.2 andconstitutes one of our more technical expositions.The importance of a meticulous handle on the various parameters appearing in our esti-mates cannot be understated. This fact justifies the rather notation-heavy nature of our writ-ing in addition to our re-exposition of the robust and powerful arguments of Anantharaman-Nonnenmacher-Koch. Our principal aim is to keep the manuscript as self-contained aspossible, given the various technicalities, and providing a number (but not all) of statementsfrom [3, 2] in blackbox form seems effective in this pursuit.In closing our outline, we emphasize that much of our notation and reproduced statementsfrom [3, 2] were taken directly from the excellent lecture notes of Nonnenmacher [29], wherethe we first learned these powerful techniques introduced by Anantharaman and Nonnen-macher himself. Many of the (possibly) desired details, which are occasionally left out fromthis manuscript, can be found in these notes and the references therein by the interestedreader.1.5.2. Outline.
The progression of our paper is as follows. Section 3 provides results on themicrolocalization of ε -logarithmic modes and why the projection of quasimodes of admissi-ble widths onto logarithmic-length intervals maintains their initial semiclassical measures.Section 4 gives standard definitions for the various entropy functionals we will use. It alsoreproduces some crucial statements regarding the composition of long strings of propagatedpseudors as well as one of the main results in [3], the hyperbolic dispersive estimate. Section5 enters new territory by discussing some properties of quantum measures for ε -logarithmicmodes, particularly their behaviour under the shift action on cylinder spaces. In fact, it is inthis section that we begin to see why we address only ε -logarithmic modes rather than gen-eral quasimodes of width ε ~ | log ~ | . We also expound upon Assumption A . Section 6 gives anexposition of a microlocal version of the Entropic Uncertainty Principle. Section 7 establishesour more precise subadditivity statements with Section 8 defining our new subprincipal-typefunctionals and their properties. The final steps of our proof are given in the latter parts ofSection 8.1.6. Notation and various parameters.
As the language of this manuscript is semiclas-sical, it is natural to simply write P ( ~ ) := − ~ ∆ g and keep this notation for the remainingdiscussion. We record a list of important parameters that appear frequently throughout ouranalysis. Note that many have been given subscripts in word form for easier associations.We should emphasize that it is not our goal to reprove the results contained in the works[3, 29] but to make this rather technical manuscript more self-contained and gentler to thereader. • E > E . • E ~ are central energies attached to a family of quasimodes { Ψ ~ } ~ . • C energy is a uniform factor attached to the distance of our central energies from thefixed energy E . • ε > • α ∈ [0 ,
1) is the exponent placed on ε and corresponds to the projection of a generalquasimode of width ε ~ | log ~ | onto I ( ε α , ~ ). -LOGARITHMIC MODES 9 • γ > I ( ε, ~ ). • K is the size of our initial partition P . Later, K ≃ ζ − d for ζ a small and positiveparameter. • ε sm > P . • ε slab > χ ( n ) ( P ( ~ )) and also λ + ( ε slab ) which is the maximal expansion rate over E . We willlater set ε sm to this parameter. • δ Ehr > T δ Ehr ,ε slab , ~ = (1 − δ Ehr ) | log ~ | λ + ( ε slab ) . Note this has an effect onthe regularity of our symbols, namely if we are in S −∞ , δ Ehr and have an expansion in P S −∞ , − j + δ Ehr δ Ehr . Towards the end of our argument, we will set δ Ehr = ε slab . • δ long > ~ due to how far we go beyond Ehrenfest time T δ Ehr ,ε slab , ~ , namely C δ long | log ~ | . We will eventually choose with δ long such that2 (1 − δ Ehr ) λ + ( ε slab ) + 1 < δ − long and δ long < δ Ehr . • n ∈ N is our fixed and finite time, independent of ~ , for the dynamics of µ sc . Thisis eventually taken arbitrarily large after all the above parameters have been madesufficiently small. • ~ >
0, such that ~ ≤ h , is determined by a number of restrictions including makingthe defining quasimode estimate valid. We choose this number last and sufficientlysmall, depending on most of the above parameters.2. Preliminaries of semiclassical calculus
In this section we recall the concepts and definitions from semiclassical analysis needed inour work. The notations are drawn from the monographies [14, 40] as well as [3].2.1.
Semiclassical calculus on M . Recall that we define on R d the following class ofsymbols for m ∈ R : S m ( R d ) := { a ∈ C ∞ ( R d × (0 , | ∂ αx ∂ βξ a | ≤ C α,β h ξ i m −| β | } . (2.1.1)Symbols in this class can be quantized through the ~ -Weyl quantization into the followingpseudodifferential operators acting on u ∈ C ∞ ( M ):Op ~ w ( a ) u ( x ) := 1(2 π ~ ) d Z R d e i ~ h x − y,ξ i a (cid:0) x + y , ξ ; ~ (cid:1) u ( y ) dydξ . (2.1.2)One can adapt this quantization procedure to the case of the phase space T ∗ M , where M is a smooth compact manifold of dimension d (without boundary). Consider a smooth atlas( f l , V l ) l =1 ,...,L of M , where each f l is a smooth diffeomorphism from V l ⊂ M to a boundedopen set W l ⊂ R d . To each f l correspond a pullback f ∗ l : C ∞ ( W l ) → C ∞ ( V l ) and a symplecticdiffeomorphism ˜ f l from T ∗ V l to T ∗ W l :˜ f l : ( x, ξ ) (cid:0) f l ( x ) , ( Df l ( x ) − ) T ξ (cid:1) . Consider now a smooth partition of unity ( ϕ l ) adapted to the previous atlas ( f l , V l ). Thatmeans P l ϕ l = 1 and ϕ l ∈ C ∞ ( V l ). Then, any observable a in C ∞ ( T ∗ M ) can be decomposed as: a = P l a l , where a l = aϕ l . Each a l belongs to C ∞ ( T ∗ V l ) and can be pushed to a function˜ a l = ( ˜ f − l ) ∗ a l ∈ C ∞ ( T ∗ W l ). We may now define the class of symbols of order m on T ∗ M (after slightly abusing notation and treating ( x, ξ ) as coordinates on T ∗ W l ) S m ( T ∗ M ) := { a ∈ C ∞ ( T ∗ M × (0 , a = X l a l , such that ˜ a l ∈ S m ( R d ) for each l } . This class is independent of the choice of atlas or smooth partition. For any a ∈ S m ( T ∗ M ),one can associate to each component ˜ a l ∈ S m ( R d ) its Weyl quantization Op ~ w (˜ a l ), whichacts on functions on R d . To get back to operators acting on M , we consider smooth cutoffs ψ l ∈ C ∞ c ( V l ) such that ψ l = 1 close to the support of ϕ l , and define the operator:Op ~ ( a ) u := X l ψ l × (cid:0) f ∗ l Op w ~ (˜ a l )( f − l ) ∗ (cid:1) ( ψ l × u ) , u ∈ C ∞ ( M ) . (2.1.3)This quantization procedure maps (modulo smoothing operators with seminorms O ( ~ ∞ ))symbols a ∈ S m ( T ∗ M ) onto the space Ψ m ~ ( M ) of semiclassical pseudodifferential operatorsof order m . The dependence in the cutoffs ϕ l and ψ l only appears at order ~ Ψ m − ~ [40,Thm 9.10]), so that the principal symbol map σ : Ψ m ~ ( M ) → S m ( T ∗ M ) / ~ S m − ( T ∗ M ) isintrinsically defined. Most of the rules and microlocal properties (for example the composi-tion of operators, the Egorov and Calder´on-Vaillancourt Theorems) that hold on R d can beextended to the manifold case.An important example of a pseudodifferential operator is the semiclassical Laplace-Beltramioperator P ( ~ ) = − ~ ∆ g . In local coordinates ( x ; ξ ) on T ∗ M , the operator can be expressedas Op wh (cid:0) | ξ | g + ~ ( P j b j ( x ) ξ j + c ( x )) + ~ d ( x ) (cid:1) for some functions b j , c, d on M . In particular,its semiclassical principal symbol is the function | ξ | g ∈ S ( T ∗ M ).We will need to consider a slightly more general class of symbols than the class (2.1.1).Following [14], for any 0 ≤ ν < / S m,kν ( R d ) := { a ∈ C ∞ ( R d × (0 , | ∂ αx ∂ βξ a | ≤ C α,β ~ − k − ν | α + β | h ξ i m −| β | } . These symbols are allowed to oscillate more strongly when ~ →
0. All the previous remarksregarding the case of δ = 0 transfer over in a straightforward manner. This slightly “exotic”class of symbols can be adapted on T ∗ M as well. For more details, see [40, Section 14.2].2.2. Anisotropic calculus and sharp energy cutoffs.
We will also need to quantizeobservables which are “very singular” along certain directions, away from some specificsubmanifold, `a la the calculus introduced by Sj¨ostrand-Zworski [35]. Let Σ ⊂ T ∗ M be acompact co-isotropic manifold of dimension 2 d − D (where D ≤ d . Near each point ρ ∈ Σ,there exists local canonical coordinates ( y i , η i ) such thatΣ = { η = η = · · · = η D = 0 } . For some index δ ∈ [0 , a ∈ S m,k Σ ,δ ( T ∗ M ) ⊂ C ∞ ( T ∗ M × (0 , • for any family of smooth vector fields V , . . . , V l tangent to Σ and of smooth vectorfields W , . . . , W l , we have in any neighborhood Σ ε of Σ:sup ρ ∈ Σ ε | V . . . V l W . . . W l a ( ρ ) | ≤ C ~ − k − l , -LOGARITHMIC MODES 11 • away from Σ, we require that | ∂ αx ∂ βξ a | = O ( ~ − k h ξ i m −| β | ).These symbols a can be split into components each supported in some adapted coordinatechart mapping to Σ ε per standard procedures, with one piece a ∞ ∈ S m,k ( T ∗ M ) vanishingnear Σ. The Weyl quantization along with the use of zeroth-order Fourier integral opera-tors associated to canonical graphs allows for a global quantization procedure. We call theresulting class Ψ m,k Σ ,δ ( T ∗ M ).We now set Σ = E . The described anistropic calculus (with respect to E ) of pseudors willplay a role mostly in the hyperbolic dispersive estimate (see Section 4.3) and the microlo-calization properties of our modes Ψ ~ (see Section 3.2). To set up the relevant pseudors, wefirst introduce some important cutoffs.Let δ long > n ≤ C δ | log ~ | where C δ + 1 < δ − . Define χ δ long ( s ) equal to 1for | s | ≤ e − δ long / and 0 for | s | ≥
1. From here we set χ ( n ) ( s, ~ ) := χ δ long ( e − nδ long h − δ long s ) . The symbols χ ( n ) ∈ S −∞ , E , − δ and they will be quantized into Op E , ~ ( χ ( n ) ( p − E ). Lemma (Disjoint supports) . (cf [29, Lemma 2.5] ) Let δ long > be given. We have dist (cid:0) Supp χ ( n ) , Supp (1 − χ ( n +1) ) (cid:1) ≥ h − δ long e δ long n ( e δ long / − . Thus, for any symbol a ∈ S m, E , − δ long and any ≤ n ≤ C δ | log ~ | , one has (cid:0) Id − Op E , ~ (cid:0) χ ( n +1) ◦ ( p − E ) (cid:1)(cid:1) Op E , ~ ( f ) Op E , ~ (cid:0) χ ( n ) ◦ ( p − E ) (cid:1) = O ( ~ ∞ ) . The same property holds if we use χ ( n ) ( P ( ~ ) − E ) . This lemma follows immediately from a standard elliptic parametrix-type argument ap-plied to the anisotropic calculus. We emphasize that one can also use functional calculus fromthe spectral theory of self-adjoint operators to define χ ( n ) ( P ( ~ ) − E ) without any referenceto pseudors. Thus, the same property in the above lemma holds if we use χ ( n ) ( P ( ~ ) − E ). Itwill prove advantageous at points to use the anisotropic calculus at times and the functionalcalculus at others, the latter especially when dealing directly with the spectrum of P ( ~ ).3. Projections, microlocalization, and differences of measures
Recall that on compact (
M, g ), the spectrum of P ( ~ ) = − h ∆ g is discrete and infinite,which we denote by { E j ( ~ ) } j ∈ N . By the spectral theorem, we can write any function Ψ ~ in L ( M ) as a linear combination of the orthonormal basis of eigenfunctions which we writeindividually as ψ j ( ~ ). In particular, if Ψ ~ is L -normalized, thenΨ ~ = X j c j ( ~ ) ψ j ( ~ )where P j c j ( ~ ) = 1. Hence, each state Ψ ~ provides a probability measure m ~ on the set ofeigenvalues { E j ( ~ ) } j ∈ N ; in particular, m ~ = X j c j ( ~ ) δ E j ( ~ ) . We can use this measure m ~ to deduce some potentially useful properties on the coefficients c j ( ~ ) for modes { Ψ ~ } ~ . We now set the notation I γ ( ε, ~ ) := (cid:20) E h − ε ~ | log ~ | − γ , E h + ε ~ | log ~ | − γ (cid:21) and introduce some important definitions.3.1. On projecting modes.
Lemma
Let Π I be the spectral projector onto I ⊂ spec P ( ~ ) where I = I γ ( ε, ~ ) , { Ψ ~ } ~ be a family of quasimodes of width ε ~ | log ~ | with central energies { E ~ } ~ . Then for any γ ∈ [0 , and the same h ( ε ) > such that for all ~ ≤ ~ the quasimode property holds, we also havethat k Π I ∁ Ψ ~ k ≤ | log ~ | − γ . Thanks to continuity of Π I on L ( M ) , { I Ψ ~ } ~ is also a family of quasimodes of width ε ~ | log ~ | with central energies { E ~ } ~ (up to renormalization by a factor of size O ( | log ~ | − γ ) ).Proof. This lemma follows from an application of Markov’s Inequality. More specifically, wehave that k X j ( E j − E ~ ) c j ψ j k ≥ s X E j ∈ I γ | c j | | E j − E ~ | ≥ s m ~ (cid:26) j | | E j − E ~ | ≥ ε ~ | log ~ | − γ (cid:27) × ε ~ | log ~ | − γ . Now apply the ε -logarithmic property on Ψ ~ after using P ( ~ ) ψ j = E j ψ j . (cid:3) Thus we are in a position to state a major reduction:
Corollary [Small differences of semiclassical measures after projection] Given
E > , γ ∈ (0 , , and a family of quasimodes { Ψ ~ } ~ with central energies { E ~ } of width ε ~ | log ~ | with a corresponding semiclassical measure ν sc , we have that the corresponding semiclassicalmeasure of the renormalized projected family of modes { c Π I Ψ ~ } ~ whose spectral support isin [ E h − ε ~ | log ~ | − γ , E h + ε ~ | log ~ | − γ ] equals µ sc .More generally, given α ∈ [0 , , there exists an ε α -logarithmic family whose semiclassicalmeasure µ sc satisfies the property that the total variation of µ sc − ν sc is less than ε − α for ε sufficiently small.Proof. Combining standard pseudor bounds with the estimate k Ψ h − c n E j ( ~ ) | | E ~ − E j ( ~ ) |≤ ε ~ | log ~ | − γ o Ψ ~ k ≤ | log ~ | − γ where c is a normalization factor of order 1 + O ( | log ~ | − γ ), shows that the above quantitygoes to 0 in ~ . Hence, for any a ∈ S −∞ , , the difference of matrix elements h Op ~ ( a )Ψ h , Ψ h i − h Op ~ ( a ) c · Π I Ψ h , c · Π I Ψ h i = h Op ~ ( a )Ψ h , Ψ h i − h Op ~ ( a ) c · Π I Ψ h , c · Π I Ψ h i + h Op ~ ( a ) c · Π I Ψ h , Ψ h i − h Op ~ ( a ) c · Π I Ψ h , Ψ h i = O ( | log ~ | − γ ) -LOGARITHMIC MODES 13 after utilizing the bilinearity of our form.A similar argument allows us to deduce, in the case of projections onto the intervals I ( ε α , ~ ), that h Op ~ ( a )Ψ h , Ψ h i − h Op ~ ( a ) c · Π I Ψ h , c · Π I Ψ h i = h Op ~ ( a )Ψ h , Ψ h i − h Op ~ ( a ) Π I Ψ h , Π I Ψ h i + h Op ~ ( a ) Π I Ψ h , Ψ h i − h Op ~ ( a ) Π I Ψ h , Ψ h ih Op ~ ( a ) Π I Ψ h , Π I Ψ h i − h Op ~ ( a ) c · Π I Ψ h , c · Π I Ψ h i + h Op ~ ( a ) c · Π I Ψ h , Π I Ψ h i− h Op ~ ( a ) c · Π I Ψ h , Π I Ψ h i = h Op ~ ( a ) (Ψ ~ − Π I Ψ h ) , Ψ h i + h Op ~ ( a ) Π I Ψ h , Ψ ~ − Π I Ψ h i + h Op ~ ( a ) (Ψ ~ − c · Π I Ψ h ) , Ψ h i− h Op ~ ( a ) c · Ψ h , c · Π I Ψ h − Π I Ψ ~ i for real symbols a such that k a k ∞ ≤ ~ ( a ) = Op ~ ( a ) ∗ . After applying adjoints,we use that the absolute value of each matrix element k Op ~ ( a ) (Ψ ~ − Π I Ψ h ) k generates apositive measure m and whose limit in ~ gives an integral R S ∗ M a dm ≤ m ( S ∗ M ). Now weapply Lemma 3.1.1 in the case of γ = 0 and I = I ( ε α , ~ ).For the terms k Op ~ ( a ) (Ψ ~ − c Π I Ψ h ) k , the normalization constant in this case takes theform c = s k Ψ ~ k − h Ψ ~ , Π I ∁ Ψ ~ i + k Π I ∁ Ψ ~ k = s − k Π I ∁ Ψ ~ k . and we keep in mind that k Π I ∁ Ψ ~ k ≤ ε − α . An application of the triangle inequality givesthat k Op ~ ( a ) (Ψ ~ − c Π I Ψ h ) k ≤ ε − α for ε small enough such that 1 − q − ε − α ≤ ε − α . (cid:3) Thus, for families of quasimodes { Ψ ~ } ~ with central energies { E ~ } of width ε ~ | log ~ | , thereexists a family of quasimodes spectrally supported in I γ ( ε, ~ ) that obtains the same semi-classical measure µ sc . And at the cost of small differences in the semiclassical measures, wecan take any such { Ψ ~ } ~ and make them into an ε α -logarithmic family.Notice that taking the strongly localizing logarithmic family described in Theorem 1.3.1,projecting it onto spectral intervals [ E h − ε ~ | log ~ | − γ , E h + ε ~ | log ~ | − γ ], and then renormalizinggives yet another strongly localizing family with width ε ~ | log ~ | but with spectral supportslightly larger than I ( ε, ~ ). Furthermore, we can choose γ arbitrarily small (but fixed).However, we will eventually see in Proposition 5.2.2 that ε -logarithmic modes appear tointeract best with the notion of shift-invariance primarily for the following reason: if { Ψ ~ } is ε -logarithmic, then k ( P − E ~ ) m Ψ ~ k ≤ ε m ~ m | log ~ | m whilst for quasimodes of width ε ~ | log ~ | projected on I γ ( ε, ~ ), we only have k ( P − E ~ ) m Ψ ~ k ≤ ε m ~ m | log ~ | m (1 − γ ) for m >
1. Thus,computing with ε -logarithmic modes is a tremendous simplification as one no longer needsto deal with the spectrum of P ( ~ ) that is “far away” from E . In other words, after invokingthe Weyl Law for manifolds admitting Anosov geodesic flows [12], we can say Corollary
Given γ ∈ (0 , , a quasimode of width ε ~ | log ~ | , from the point of viewof identical semiclassical measures, a linear combination of ∼ ε h − ( n − | log ~ | − γ eigenfunctions. Andup to small differences of semiclassical measures, they are a linear combinations of at most C M h − ( n − | log ~ | eigenfunctions where C M > comes from the Weyl remainder. Microlocalization properties.
The main statement of this section is a lemma on thewavefront set for our ε -logaritmic modes. Lemma ( ε -logarithmic modes are microlocalized at scale h − δ ) . Let { Ψ ~ } ~ be a familyof ε -logarithmic modes and consider I ( ε, ~ ) , δ ∈ (0 , , and C δ < δ − − (as in Section 2.2).Then there exists h ( ε ) such that for all ~ ≤ h and n ≤ C δ | log ~ | , we have (cid:0) Id − χ ( n ) ( P ( ~ ) − E ) (cid:1) Ψ ~ = O ( h ∞ ) . In fact, if χ ( n ) ( P ( ~ ) − E ) is defined through functional calculus, then the right-hand sideis 0. As a consequence, ε -logarithmic modes are microlocalized inside the energy layer ofwidth h − δ around E and therefore W F ~ ( ψ ~ ) ⊂ E . The same statement holds for a family ofquasimodes spectrally supported in I γ ( ε, ~ ) .Proof of Lemma 3.2.1. One can see this in two ways, both of which we will present. Thefirst involves a computation via spectral calculus after using the support properties of χ ( n ) .Thanks to χ ( n ) ( P ( ~ ) − E ) being a function of P ( ~ ), we have χ ( n ) ( P ( ~ ) − E ) Ψ = X E j ∈ I ( ε,h ) χ ( n ) ( P ( ~ ) − E ) c j ψ j where ψ j is a eigenmode with eigenvalue E j and [ E − h − δ , E + h − δ ] as dictated by thecutoff χ ( n ) ( P ( ~ ) − E ) . It is important to note that I γ ( ~ ) ⊂ [ E − h − δ , E + h − δ ] as long as ~ | log ~ | ≤ ε − . Thus, P E j ∈ I ∁ ( h ) (cid:0) − χ ( n ) ( E j − E ) (cid:1) c j ψ j = 0 as c j = 0 in this summationrange.One can allow use the second-microlocal calculus developed extensively by Sj¨ostrand-Zworski [35] and some standard arguments, as follows. Consider the eigenvalue E j ∈ I ( ~ )that is furthest from the energy E . From [3, Proposition 2.5], we know that k (cid:0) I − χ ( n ) (( P ( ~ ) − E )) (cid:1) ψ j k = O ( ~ ∞ )by constructing a left elliptic parametrix for P ( ~ ) − E ), and then utilizing a property forsymbols of disjoint support. As I ( ~ ) ⊂ [ E − h − δ long , E + h − δ long ] for h ∈ (0 , k (cid:0) I − χ ( n ) (( P ( ~ ) − E ) (cid:1) Ψ ~ k = O ( ~ ∞ ) by simply applying the triangle inequality { E j | E j ∈ I } times, keeping in mind that Ψ ~ is normalized and therefore each | c j | ≤
1, and finallyyielding to the Weyl Law. (cid:3)
We emphasize that the spectral support of χ ( n ) ( P ( ~ ) − E ) being the interval [ E − h − δ , E + h − δ ] was used crucially in our proof of microlocalization of purely logarithmic modes around E . Hence, despite having possibly varying central energies E ~ , we continue to have stronglocalization around the fixed energy E .To emphasize the advantages of using spectral calculus as well as the necessity of the WeylLaw to establish strong microlocalization, we contrast this against against the more standardlooking, but substantially weaker, microlocal estimate of (cid:0) Id − χ ( n ) ( P ( ~ ) − E ) (cid:1) Ψ ~ = O (cid:18) ~ | log ~ | (cid:19) , which holds for any ε -logarithmic mode. To see this, one could again use the second-microlocal calculus that is developed extensively in [35]. With this machinery in hand,the remaining steps are to follow a standard elliptic parametrix argument resulting in an -LOGARITHMIC MODES 15 operator (( P ( ~ ) − E ) + iQ ( ~ )) − , use the ε -log mode estimate along with ( I − χ ( n ) )( P ( ~ ) − E ) + iQ ( ~ ) − ( P ( ~ ) − E ~ ) = ( I − χ ( n ) )( P ( ~ ) − E ) + O ( ~ ∞ ) thanks to the microlocal supportof Q ( ~ ). Clearly, this estimate does not establish microlocalization as we need ~ ∞ errors. Remark . The proof ofthis lemma shows that for a family of modes Ψ ~ that are the sum of eigenfunctions whoseeigenvalues are inside of a fixed (possibly ~ -dependent) interval I , then W F ~ ⊂ ∪ E ∈ I S ∗ E M ,an energy slab. 4. Entropy and Quasiprojectors
Definitions: Entropy and quasiprojectors.
The material in this section is takenfrom [29, Section 3] and whose notation has been maintained for smoother referencing. Wereproduce the major points for easier reading and leave the minor details to the interestedreader.4.1.1.
Classical entropy and pressure.
For more information on entropy for dynamical sys-tems, see the classical texts of [23] and [7, Chapter 9].
Definition
Let µ be a g t -invariant probability measure on E and P = ( E , . . . , E K )be a µ -measurable partition on E . We say the metric entropy of µ with respect to P is thequantity H ( µ, P ) := K X k =1 η ( µ ( E k )) where η ( s ) := − s log s and s ∈ [0 , . We should emphasize that as − log( s + s ′ ) − ≤ − log( s ) − s ′ ∈ [0 , | η ( s + s ′ ) − η ( s ) | ≤ η ( | s ′ | ) by the fundamental theorem of calculus. This will proveto be useful in later sections.In order to compute what is referred to as the Kolmogorov-Sinai entropy of µ , we firstneed to consider dynamical refinements P ∨ n (or alternatively written as [ P ] n − ) where n ≥ E α := g − ( n − E α n − ∩ . . . g − E α ∩ E α where α := α . . . α n − is a sequence/word of length n with symbols α k ∈ { , . . . , K } . Notethat many such E α will be empty however we will continue to sum over all possible wordsof length n when computing the associated metric entropies. Moreover, we write [ P ] m + n − m to be the refinement with elements g − m E α := g − m − ( n − E α n − ∩ . . . g − m − E α ∩ g − m E α . With all this notation in place, if the measure µ and initial partition P are unambiguous,we will simply write H m + n − m ( µ ) := H (cid:0) µ, [ P ] m + n − m (cid:1) . Definition
Given an invariant probability measure µ , the Kolmogorov-Sinai entropy H KS ( µ ) is the number H KS ( µ ) := sup P lim n →∞ n H n − ( µ ) = sup P lim n →∞ n H n − − n ( µ ) . For an Anosov flow, we can always guarantee the existence of a partition P that is agenerator for the flow, and hence this supremum will be obtained by P [7, Theorem 9.3.5].4.1.2. Classical pressure.
Given our probability measure µ and a partition P = ( E , . . . , E K ),we can associate a set of weights w = { w i > k = 1 , . . . , K } to this pairing. This leads tothe notion of pressure. Definition
Given a probability µ , a partition P = ( E , . . . , E K ), and weights w = { w i > k = 1 , . . . , K } , we define the associated pressure as p ( µ, P , w ) := − K X k =1 µ ( E k ) log (cid:0) w k µ ( E k ) (cid:1) . This bears the question of how to define our weights when the partition P is refined viathe dynamics of g t . Thus, we introduce the next definition: Definition
Given a set of weights w = { w i > k = 1 , . . . , K } associated to apartition P , we define the n-th refinement of w as the set consisting of the elements w α = n − Y j =0 w α j , | α | = n where w α is attached to the element E α ∈ P ∨ n . Thus, we can write the pressure associatedto these refined weights as p n − ( µ, P , w ).One can show the pressure p n − satisfies similar subadditivity properties as the metricentropy H n − .4.1.3. Entropy of smooth partitions.
Let P = ( E , . . . , E K ) be an initial partition of E con-sisting of µ sc -measurable sets. Let k be the characteristic function on E k and define α := α n − ◦ g n − × · · · × α ◦ g × α (4.1.5)which is the characteristic function of E α . Thus for any measure µ on E , we have µ ( E α ) = R E α dµ .Thanks to the construction of special partitions P carried out in [1, Appendix A.2], wecan assume µ sc does not charge the boundaries of our initial partition elements E k . Thisleads us to our next definition: Definition
Given any ε sm > P = ( E , . . . , E K ) of E , there exists afamily of functions { π k } Kk =1 and neighbourhoods { ˜ E k } k with the following properties: π k ∈ C ∞ ( E ε sm , [0 , π k ⊆ ˜ E k , E k ⊆ ˜ E k , and K X k =1 π k ( ρ ) = χ ε sm / , Supp χ ε sm / ⊂ E ε sm , and χ ε sm / = 1 on E ε sm / . (4.1.7)We call such a family { π k } Kk =1 a smooth partition of unity near E . -LOGARITHMIC MODES 17 We can extend the definition of entropy to smooth partitions P sm = { π k } k as follows.First, let µ ( π k ) := Z E εsm π k dµ. Here, we have extended our definition of µ from E to E ε sm by writing µ := C d r d − ε sm dr ⊗ dµ ( ρ )where ( r, ρ ) ∈ (1 − ε sm , ε sm ) × E , with C d > n and used simply to make the extended µ a probabilitymeasure. Here ( r, ρ ) are the polar coordinates for a point ( x ; ξ ) ∈ T ∗ M where r = | ξ | g .Notice that by Lebesgue differentiation, as ε →
0, we return our initial semiclassical measure µ sc . Without loss of generality, let us assume C d = 1. Notice that µ ε is ( Id × g t )-invariantprobability measure. We are now in a position to give a rigorous definition of entropy forsmooth partitions: Definition
Given a smooth partition P sm , associated weights w , and a small pa-rameter ε >
0, we define the entropy of a smooth partition P sm as H ( µ, P sm ) := − K X k =1 µ ( π k ) log ( µ ( π k )) . (4.1.9)and p ( µ, P sm , w ) := − K X k =1 µ ( π k ) log (cid:0) w k µ ( π k ) (cid:1) . (4.1.10)The corresponding definitions of P ∨ n sm , H n − , and p n − are clear.Thanks to this setup and our assumption that µ sc does not charge the boundaries ∂E k forall k = 1 , . . . , K , we have that given any ε ′ > n ≥
1, we can choose ε sm small enoughand therefore a corresponding P sm such that | H ( µ, P ∨ n sm ) − H ( µ, P ∨ n ) | ≤ ε ′ where the elements of P ∨ n sm follow the formula given in (4.1.5). This follows easily fromcontinuity properties of − x log( x ) along with the formula for the extension of µ sc and whoseproof is left to the interested reader.4.1.4. Quasiprojectors.
Analogous to our smooth partition of unity { π k } Kk =1 with properties(4.1.7), we can form a quantum partitions of unity: Definition
A(n) (approximate) quantum partition of unity is a collection P sm , q = { Π k := Op ~ (˜ π k ) } k , whose individual elements Π k we call quasiprojectors with correspondingsymbols ˜ π k ∈ S −∞ , ( T ∗ M ) that satisfy the following properties:(1) for each k , the symbol ˜ π k is real with Supp ˜ π k ⊂ ˜ E k and admits √ π k as a principalsymbol, making Π k self-adjoint. (2) the family of operators P sm , q satisfies the relation K X k =1 Π k = Op ~ ( ˜ χ ε sm / ) + O ( ~ ∞ ) (4.1.12)where ˜ χ ε sm / ∈ S −∞ , ( T ∗ M ) itself satisfies k Op ~ ( ˜ χ ε sm / ) k = 1 + O ( ~ ∞ ) , Supp χ ε sm / ⊂ E ε sm , and χ ε sm / = 1 on E ε sm / . Naturally, we have that σ (Op ~ ( ˜ χ ε/ )) = χ ε/ . Notice that Π k cannot simply be Op ~ ( √ π k )as we need lower order symbols in the expansion in order to guarantee property (4.1.12);otherwise, our error term would be O ( h ). The parameter ε sm will become ε slab later on inthe manuscript.4.1.5. Refined quantum partitions.
Analogous to the refinements P ∨ n sm of an initial smoothpartition P sm , we can consider their quantum versions. Definition
Consider the Schr¨odinger propagator U = e − iP ( ~ ) / ~ . Then for a se-quence α of length n ∈ N , we define the refined (forward) quasiprojectors asΠ α := U − ( n − Π α n − U Π α n − . . . Π α U Π α The family P ∨ nsm,q = { Π α } | α | = n is called the n -th refinement of the quantum partition P sm , q .It is naturally that our refined quasiprojectors continue to form quantum partitions ofunity, as encompassed in the next proposition. Proposition (Refined quasiprojectors form a(n) (approximate) quantum partitionof unity) . (cf [29, Prop 3.1] ) Take n max = (cid:2) C δ long | log ~ | (cid:3) . Then for each ≤ n ≤ n max, therefined quantum partition P ∨ nsm,q continues to form a quantum partition of unity microlocallynear E , in the following sense. For any symbol χ ∈ S −∞ , ( T ∗ X ) supported inside of E ε/ , wehave ∀ n, X α Π ∗ α Π α = S n , k S n k = 1 + O ( ~ ∞ )( Id − S n ) Op ~ ( χ ε/ ) = O ( ~ ∞ )4.2. Long-time Egorov and compositions.
Definition
Let δ Ehr , ε slab > , and ~ > T δ Ehr ,ε slab , ~ := (1 − δ Ehr )2 λ + ( ε slab ) | log ~ | (4.2.2)the Ehrenphest time on the energy neighborhood E ε slab . The number λ + ( ε slab ) is maximalexpansion rate across the entire slab E ε slab . Proposition (Long-time Egorov) . Fix δ Ehr > , ε slab > , and ν ∈ ((1 − δ Ehr ) / , / .Take a ∈ S −∞ , , supported inside of E ε . Then, for any ~ ∈ (0 , and any time t = t ( ~ ) inthe range | t | ≤ T δ Ehr ,ε slab , ~ , we have U − t Op W ~ ( a ) U t = Op W ~ (˜ a t ) + O ( ~ ∞ ) where a t − ˜ a t ∈ S −∞ , − (1+ δ Ehr ) / ν and a t ∈ S −∞ , ν . -LOGARITHMIC MODES 19 Remark . In Proposition 4.1.14, we can go “long” beyond T ε,δ Ehr , ~ (hence the use of δ long ) and stay in a good symbol calculus. The reason is because we only care about very softproperties of the quasiprojectors, rather than a precise description on their pseudodifferentialstructure.The following is taken from [30, Theorem 7.1]: Proposition (Refined quasiprojectors form a bounded set of pseudors) . Take δ Ehr ∈ (0 , and ν ∈ [(1 − δ Ehr ) / , / . Then the family of symbols { ˜ π α , | α | ≤ T δ Ehr ,ε slab , ~ } belongsto a bounded set in the class S −∞ , ν . Furthermore, the refined quasiprojectors satisfy Π α − Op W ~ (˜ π α ) ∈ Ψ −∞ , − (1+ δ ) / ν Corollary
Take δ Ehr ∈ (0 , and ν ∈ [(1 − δ Ehr ) / , / . Let α, β be two sequencesof length n ≤ T ε, ~ . Then the symbols ˜ π α , ˜ π β ◦ g − n belong to S −∞ , ν , and so does their product.The operator Π β × α := Π α U n Π β U − n equals U n Π βα U − n , belongs to Ψ −∞ , ν , and satisfies Π β × α = Op W ~ (cid:0) ˜ π α × ˜ π β ◦ g − n (cid:1) + Ψ −∞ , − νν . Remark . We gather from these statements that for n ≤ ⌊ T δ Ehr ,ε shell , ~ ⌋ , certain com-positions of quasiprojectors are true quantizations of refined, smooth partitions elementsin P ∨ n sm . Using that U n Π βα U − n = Π β × α , we obtain the same operator bound for Π βα thanks to unitarity. Thus, for (2-sided) dynamical sequences α of length ≤ T ε, ~ , we have k Π α k = k ˜ π α k ∞ + O ( h − ν ).4.3. Hyperbolic dispersive estimates.
Definition (Coarse-grained unstable Jacobian) . Let P sm = ( π , . . . , π K ) be a smoothpartition of E ε sm / and consider the refinement P ∨ n sm . For E α j ∈ P ∨ n sm , we define J un ( α j ) :=inf { J u ( ρ ) | ρ ∈ ˜ E α j } . From this, we call the quantity J un ( α ) := n − Y i =0 J u ( α i ) (4.3.2)the coarse-grained unstable Jacobian at α . Proposition (Hyperbolic dispersive estimate) . Let δ Ehr and ε small be given. Choose δ long > small enough with λ max + 1 < δ − long . Furthermore, consider a partition P . Then,there exists ~ ( P , δ long , δ Ehr ) > , C > , and c > such that for any < ~ ≤ ~ , anyintegers n, m ∈ [0 , T δ Ehr ,ε slab , ~ ] and any two sequences α , β of length n , the following estimateholds: k Π ∗ α U n Π β U − n χ ( n ) ( P ( ~ )) k ≤ C h − ( d − c δ long ) / J un ( α ) − / J un ( β ) − / (4.3.4)Let Λ u min be the minimal ( d − M . We remindthe reader that given ε ′ >
0, for all n and α such that | α | = n , there exists C > J un ( α ) − ≤ Ce − n (Λ u min − ε ′ ) . Thanks to these bounds on the coarse-grainedJacobians, the hyperbolic estimate improves past the trivial pseudor bound for times n ≥ ( d − | log ~ | Λ u min > T δ,ε, ~ The entropic uncertainty principle (described in Section 6.2) imposes that we have theparticular estimate (4.3.4). Moreover, we see that n > T δ,ε, ~ is necessary in order to havea product-of-quasiprojectors polynomial-type decay bound; this comes immediately aftercalculating the minimal and maximal expansion rates for the Anosov flow. More specifically,we take the following sequence of steps: 1) Take a product of pseudors of length n = ⌊ T δ Ehr ,ε slab , ~ ⌋ , one associated to positive times and the other to negative times; this is a goodpseudor, 2) Conjugate this pseudor by U n for the same n to get another operator (maynot be a pseudor anymore) associated to a forward sequence of length 2 n , 3) Take anothersuch two-sided pseudor and conujgate that by U − n to get another operator associated to abackwards sequence of length 2 n , 4) take their product to get a two-sided sequence of fulllength 4 n (as in (4.3.4), and finally 5) remember that you only need a norm estimate in theend.In terms of keeping your compositions within a good pseudor class, notice that we coulditerate this process but are limited to conjugation by U n ′ where n ′ ≤ ⌊ T δ ′ ehr ,ε slab , ~ ⌋ where δ ′ Ehr + δ Ehr < S , −∞ δ ′ Ehr + δ Ehr . Therefore, one could considera sequence of conjugations at times below the Ehrenfest time, say at n j = ⌊ T δEhr j ,ε slab , ~ ⌋ for j ≥ P Nj =0 n j where you start composing from n N and apply longer and longer conjugation times6. This is well past twice the Ehrenfest time.The following insightful, yet natural, interpretation of Proposition 4.3.3 is taken from [29]and reproduced here for the interest of the reader: Remark . For times n > T δ Ehr ,ε slab , ~ , the measure µ h is necessarily distributed over many cylinders of length n .This corresponds to a dispersion phenomenon: the state Ψ ~ cannot be concentrated in O (1)boxes E α , since each such box has a volume of ~ d − . This follows immediately from the L -operator norm of a pseudor being bounded above by the L -norm of the its symbol overphase space along with the dispersive estimate.5. Quantum measures
Definitions: Quantum symbolic measures.
Thanks to the results in Section 3.2,we know that ε -logarithmic families { Ψ ~ } ~ have P α = n k Π α Ψ ~ k = 1 + O ( h ∞ ) as long as n ≤ n max as defined in Proposition 4.1.14. Hence, up to O ( h ∞ ) errors, we have a notionfor a probability measure on the symbolic space on n -cylinders Σ n . This leads to our nextdefinition Definition (Quantum symbolic measures) . Let { Ψ ~ } ~ be an ε -logarithmic family.Then for any α ∈ Σ n with n ≤ n max , we write µ ~ ([ · α ]) := k Π α Ψ ~ k , (5.1.2)which itself we call the forward (quantum) symbolic measure . Analogously, we can define the backwards (quantum) symbolic measure for { Ψ ~ } ~ as˜ µ ~ ([ · α ]) := k Π α n − ( − n ) . . . Π α ( − ~ k , More formally, µ ~ and ˜ µ ~ can be seen as an equivalence classes of ~ -dependent probabilitymeasures on n -cylinders with the equalities being true modulo O ( h ∞ ) quantities. Moreover, -LOGARITHMIC MODES 21 for n fixed, µ ~ ([ · α ]) → µ sc ( π α ) as ~ → µ ~ but nowcorresponding to backwards elements π α ∈ P ∨− n sm . Note that for both measures, our word α is read from left to right with its corresponding quasiprojector being written from right toleft.We conclude this section with lemma on the structure of our probability measures: Lemma (Compatibility conditions for logarithmic modes) . Let { Ψ ~ } ~ be a family ofquasimodes with spectral support in I γ ( ε, ~ ) and δ Ehr , ε slab > be given. Then there exists ~ such that for ~ ≤ h and for all n ≤ ⌊ T δ Ehr ,ε slab , ~ ⌋ we have , X | α | = n k Π α Ψ ~ k = 1 + O ( ~ ∞ ) , and ∀ α = ( α , . . . , α n − ) , k Π α ,...,α n − Ψ ~ k = X α n − k Π α ,...,α n − Ψ ~ k + O ( h ∞ ) Proof.
This is an immediate implication of Proposition 4.1.14 and Lemma 3.2.1. (cid:3)
Shift invariance.
Definition (Shift action on a quantum measure) . For any symbolic measure asso-ciated to purely logarithmic family { Ψ ~ } ~ , we define the action of the shift map σ on thesymbolic space of n -cylinders Σ n as follows: µ ~ (cid:0) σ − n [ · β ] (cid:1) := X | α | = n (cid:13)(cid:13) Π β ( n ) (cid:2) Π α n − ( n − . . . Π α (0) (cid:3) Ψ ~ (cid:13)(cid:13) and ˜ µ ~ ( σ n [ β · ]) := X | α | = n (cid:13)(cid:13) Π β ( − n ) (cid:2) Π α n − ( − ( n − . . . Π α (0) (cid:3) Ψ ~ (cid:13)(cid:13) where | β | = n , Π β ( n ) = U − n Π β U n , and Π β ( − n ) = U n Π β U − n . We emphasize that for the forward symbolic measure, we are “adding n letters to the left”in the forward symbolic space and then summing over the interior partition elements andsimilarly for the backwards measure. Similarly to [3, Proposition 4.1] and [29, Lemma 3.14],we have a more general statement for the σ -invariance of our quantum measures: Proposition (Approximate shift-invariance for logarithmic modes) . Let µ ~ , ˜ µ ~ be theforward/backward quantum symbolic measures associated with a family { Ψ ~ } ~ of ε -logarithmicmodes. Fix some n ≥ , δ Ehr , and ε small . Then, there exists ~ ( n ) such that for all ~ ≤ ~ ( n ) , any β of length n , and n in the range [0 , T δ Ehr ,ε small , ~ − n ] we have µ ~ (cid:0) σ − n [ · β ] (cid:1) = µ ~ ([ · β ]) + F ( ~ , β , ε, n ) , ˜ µ ~ ( σ n [ · β ]) = ˜ µ ~ ([ · β ]) + ˜ F ( ~ , β , ε, n ) (5.2.3) where F ( ~ , β , ε, n ) = − n ~ Im h Π β ( P ( ~ ) − E ~ )Ψ ~ , Π β Ψ ~ i (5.2.4)+ ∞ X m =2 i m m ! ~ m m X l =0 (cid:18) ml (cid:19) n m h Π β ( P − E ~ ) l Ψ ~ , Π β ( E ~ − P ) m − l Ψ ~ i + O n ( h − ε ′ ) and ˜ F is the same but with each instance of Π β replaced by the backwards quasiprojector e Π β , for some ε ′ > . It is important to note that for quasimodes with spectral support in I γ ( ε, ~ ), we havethe estimate h Π β ( P − E ~ ) l Ψ ~ , Π β ( E ~ − P ) m − l Ψ ~ i ≤ ε m ~ m | log ~ | m (1 − γ ) as we do not have moreexplicit information for the coefficient corresponding to energies inside outside of I ( ε, ~ ).Therefore for ε -logarithmic modes, we have an ~ -independent upperbound on the functions F ( ~ , β, ε, n ). This result should be contrasted against the shift-invariance statement inAnantharaman’s work [1, Proposition 1.3.1] and Nonnenmacher’s exposition [29, Lemma3.14]. Proof.
We proceed almost exactly the same as in [29, Lemma 3.14] until the last few steps.We will do the calculation for the forward measure first. We have X α ∈A n h| Π β ( n ) | Π α Ψ ~ , Π α Ψ ~ i = X α ∈A n X α n − (cid:0) h Π α n − ( n − ∗ | Π β ( n ) | Π α n − ( n − α ′ Ψ ~ , Π α ′ Ψ ~ i (cid:1) = X α ′ X α n − h Π α n − ( n − ∗ Π α n − ( n − | Π β ( n ) | Π α ′ Ψ ~ , Π α ′ Ψ ~ i + X α ′ X α n − h Π α n − ( n − ∗ (cid:2) | Π β ( n ) | , Π α n − ( n − (cid:3) Π α ′ Ψ ~ , Π α ′ Ψ ~ i = X α ′ ∈A n − h| Π β ( n ) | Π α ′ Ψ ~ , Π α ′ Ψ ~ i ! + O n ( h − ν ) · X α n − ∈A k Π α n − k + K n − · O ( ~ ∞ ) + K · O ( ~ ∞ )The final line is deduced as follows. First, we use that that P α n − Π α n − ( n − α n − ( n − ∗ = Op ~ ( χ ε/ ) + O ( h ∞ ), thanks to Definition 4.1.12, and then apply the strong microlo-calization property for p.l. modes. This is returns the term of K · O ( ~ ∞ ). Now let ν < / (cid:2) | Π β ( n ) | , Π α n − ( n − (cid:3) = O n ( ~ − ν ), sum first in α n − in order gain theterm P α n − ∈A k Π α n − k , and then use that P α ′ k Π α ′ Ψ ~ k = 1 + O ( ~ ∞ ); this leads to thefinal term of K · O ( ~ ∞ ). The assumption that n + n ≤ ⌊ T δ Ehr ,ε slab , ~ ⌋ is crucial in estimat-ing our commutator (cid:2) | Π β ( n ) | , Π α n − ( n − (cid:3) ; this follows from an application of Proposition4.2.5, invariance of norms under conjugation by U − n/ , and the fact that compositions withinsingular pseudor classes are closed.We can repeat this procedure n − P α ∈A n h| Π β ( n ) | Π α Ψ ~ , Π α Ψ ~ i equals h| Π β | U n Ψ ~ , U n Ψ ~ i + n · O n ( h − ν ) · X α n − ∈A k Π α n − k + n − X j =1 K j ! · O ( ~ ∞ ) + K · n · O ( ~ ∞ ) . We pause to note that Ψ ~ is being tested with partition elements corresponding to [ P sm,q ] n − . -LOGARITHMIC MODES 23 Now we will apply the propagator to the ε -logarithmic modes and expand out to give: X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ ) e in ( E j − E k ) / ~ h Π β ψ j , Π β ψ k i . And expansion of e in ( E j − E k ) / ~ up to order 2 leads to the sum above being of the form X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ ) h Π β ψ j , Π β ψ k i + in ~ × (cid:16) X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ )( E j − E ~ ) h Π β ψ j , Π β ψ k i− X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ )( E k − E ~ ) h Π β ψ j , Π β ψ k i (cid:17) + X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ ) f ( j, k, ~ ) h Π β ψ j , Π β ψ k i where f ( j, k, ~ ) = O (cid:16) n | E j − E k | ~ (cid:17) . This expression reduces to h Π β Ψ ~ , Π β Ψ ~ i − n ~ Im h Π β ( P ( ~ ) − E ~ )Ψ ~ , Π β Ψ ~ i + X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ ) f ( j, k, ~ ) h Π β ψ j , Π β ψ k i . The absolute value of the second term is bounded above by n ~ k Π β k k ( P ( ~ ) − E )Ψ ~ k k Π β Ψ ~ k .However, the third term is problematic, since it’s not necessarily a product of functionsdepending separately on j and k .So, we will use that the exponentials are analytic. Using e in ( E j − E k ) / ~ = ∞ X m =0 i m ~ m n m ( E j − E k ) m m != ∞ X m =0 i m m ! ~ m n m m X l =0 (cid:18) ml (cid:19) ( E j − E ) l ( E − E k ) m − l an application of functional calculus to the operators ± ( P ( ~ ) − E ) gives us the sum of matrixelements X E j ,E k ∈ I c j ( ~ ) c k ( ~ ) ∞ X m =0 i m m ! ~ m n m m X l =0 (cid:18) ml (cid:19) h Π β ( P − E ) l ψ k , Π β ( E − P ) m − l ψ j i . Thanks to the absolute convergence of the sum, we interchange the sums to arrive at ∞ X m =0 i m m ! n m ~ m m X l =0 (cid:18) ml (cid:19) h Π β ( P − E ~ ) l Ψ ~ , Π β ( E ~ − P ) m − l Ψ ~ i . After a single application of Cauchy-Schwarz on each term, we can utilize the quasimodeestimate as follows: k Π β ( P − E ~ ) l Ψ ~ k = h| Π β | ( P − E ~ ) l Ψ ~ , ( P − E ~ ) l Ψ ~ i / ≤ k| Π β | ( P − E ~ ) l Ψ ~ k / · k ( P − E ~ ) l Ψ ~ k / ≤ k Π β k ε l h l | log ~ | l (1 − γ ) . Note that the commutator [Π β , ( P − E ~ )] = O ( ~ ), which dominates the actual quasimodebound, hence why we rely on na¨ıve combination of Cauchy-Schwarz with some simple oper-ator calculus.This shows that the final sum of X E j ( ~ ) ,E k ( ~ ) ∈ I γ ( ~ ) c j ( ~ ) c k ( ~ ) f ( j, k, ~ ) h Π β ψ j , Π β ψ k i is bounded above by k Π β k ∞ X m =2 i m m ! (cid:18) ε · n | log ~ | (cid:19) m = k Π β k (cid:18) e i ε · n | log ~ | − γ − − ε n | log ~ | − γ i (cid:19) for ~ ≤ ~ ( n ) and ε small enough; in particular, as ε small and δ Ehr will later be taken to 0,and n + n ≤ T ε,δ, ~ , we can set the restriction of ε < λ max ~ sum in the case of γ = 0.Note that our admissible values of ε does not depend on β .Hence, the terms after − n ~ Im h Π β ( P ( ~ ) − E ~ )Ψ ~ , Π β Ψ ~ i are now O n ( ε ), so this proves theformula for F ( ~ , β , ε, n ). In fact, it will be o n ( ε ) for ε sufficiently small but now depending β a priori. The estimate for the backwards measures follows exactly the same. (cid:3) Note that the threshold for shift-invariance is lower than the threshold for invarianceof the whole semi-classical measure: o ( ~ ), emphasizing that we are in fact utilizing a morerefined property for µ sc . Moreover, our particular method of estimation falls apart for modesspectrally supported in I γ ( ε, ~ ). If one could generalize Proposition 5.2.2 to such modes, thenCorollary 3.1.2 tells the projections of quasimodes of width ε ~ | log ~ | onto I γ ( ε, ~ ) produce thesame semiclassical measures, potentially allowing one to generalize our Theorem 1.2.1.5.3. On Assumption A.
We are in a position to properly state the assumption eluded tojust before Theorem 1.2.1:
Assumption A : Consider the hypotheses of Proposition 5.2.2 and F , ˜ F defined in (5.2.4).Given ε >
0, there there exists ~ independent of β, ε, and n and δ ( ε ) > β where µ ~ ([ · β ]) < δ when ~ ≤ ~ , we have thatsup n ≤⌊ T δEhr,εslab, ~ ⌋− n | F ( ~ , β , ε, n ) | < ε for the same range of ~ .First, notice that Assumption A is trivial when ε = o (1) in ~ . Moreover, this assump-tion is simply the limiting definition of absolulte continuity of measures (or rather, linear -LOGARITHMIC MODES 25 functionals). Our assumption appears to require a knowledge of the asymptotics of µ ~ awayfrom the support of µ sc , but contained in E , for which our current microlocal technologydoes not penetrate. However, we can provide a proof of the absolute continuity of only some“component” functionals, namely i m m ! ~ m n m h Π β ( P − E ) l ψ k , Π β ( E − P ) m − l ψ j i and which is given in Section 8.2; see the corresponding remarks for a further discussion onobstructions and possible ways forward.It is worth also mentioning that one could also write F in the following way withoutanalytic series: h Π β (cid:0) e in ( P − E ~ ) / ~ − (cid:1) Ψ ~ , Π β (cid:0) e in ( E ~ − P ) / ~ − (cid:1) Ψ ~ i− h Π β e in ( P − E ~ ) / ~ Ψ ~ , Π β Ψ ~ i + h Π β Ψ ~ , Π β e in ( E ~ − P ) / ~ Ψ ~ i . This expression potentially provides a simple way of using functional calculus and possibly afruitful application of commutators possibly through the Baker-Campbell-Hausdorff formula.6.
Quantum entropies and the entropic uncertainty principle
Definitions: Quantum entropies and pressures.
The analogies with classical pres-sures and entropies defined in Section 4.1 are rather straightforward at this point.
Definition
Given an L -normalized quantum state Ψ ~ , a smooth quantum partition P sm , q , and an associated set of weights w , can define the quantum entropy as H n − (Ψ ~ , P sm , q ) := H n − ( µ ~ )and the quantum pressure as p n − (Ψ ~ , P sm , q , w ) := p n − ( µ ~ , w )as in Section 4.1.We pause and reproduce another insightful remark from [29] on an intuitive applicationof Proposition 4.3.3 on H n − : Remark . Theestimate in Proposition 4.3.3 on the weights of “long” cylinders has a direct consequence onthe quantum entropies: H n − (Ψ ~ , P sm,q ) ≥ n Λ u min − ( d − cδ ) | log ~ | − log C for n > T ≥ ⌊ T δ Ehr ,ε slab , ~ ⌋ , in the case of eigenmodes and purely logarithmic modes.It looks promising to then deduce h KS ( µ sc ) ≥ Λ u min . However, there are two issues: 1)a relatively minor one being that µ ~ is not probability and therefore we will need to trackerror terms in order to properly utilize the upper semi-continuity of h KS , but more seriously2) for n > T , the matrix elements µ ~ ([ · β ]) don’t necessarily converge to µ sc ( E β ) due thelong-time Egorov Theorem breaking down. Hence our symbolic probability measures cannotbe shown to converge to actual probability measures on T ∗ M . (cid:3) Uncertainty principles.
The following proposition is a recording of [29, Proposition3.12] and forms a crucial component of the final stages of the proof of Theorem 1.2.1. Wekeep the more abstract form of the statement for a cleaner exposition.
Proposition (Entropic uncertainty principle, microlocal weighted partitions) . On aHilbert space H , consider two (approximate) quantum partitions of unity of bounded operators ρ = { ρ i , i ∈ I } and τ = { τ j , j ∈ J } satisfying the identities X i ∈ I ρ ∗ i ρ i = S ρ , X j ∈ J τ ∗ j τ j = S τ , and two families of weights v = { v i , i ∈ I } , w = { w j , j ∈ J } satisfying V − ≤ v i , w j ≤ V for some V ≥ .Assume that for some ≤ W ≤ min {| I | − V − , | J | − V − } , the above sum of operatorssatisfies ≤ k S ρ k op , k S τ k op ≤ W . Furthermore, let S c , S c be two Hermitian operatorson H satisfying ≤ k S c k op , k S c k op ≤ W and the relations k ( S c − Id) ρ i S c ) k op ≤ W , for i ∈ I k ( S ρ − Id) S c ) k op ≤ W , and k ( S τ − Id) S c ) k op ≤ W . Let us set c cone := max i ∈ I,j ∈ J v i w j k ρ i τ j S c k op . Then for any Ψ ∈ H normalized such that k (Id − S c ) k op ≤ W , the quantum pressureswith respect to the weighted partitions ( ρ , v ) and ( τ , w ) satisfy the bound p (Ψ , ρ , v ) + p (Ψ , τ , w ) ≤ − (cid:0) c cone + 3 | I | V W (cid:1) + O (cid:0) W / (cid:1) with the implicit constant in the big-O notation being independent of the weighted partitionsor cutoff operators S c , S c . The formula for c cone determines our choice of quantum partitions and weights; in par-ticular, we need that τ j ρ ∗ i S c be equal to Π β × α χ ( n ) ( P ( ~ ) − E ) (see Corollary 4.2.6) where | α | = | β | = n and n = ⌊ T δ Ehr ,ε slab , ~ ⌋ . The following list sets the remaining choices in place: τ = { Π α | | α | = n } ρ = { U n Π ∗ β U − n | | β | = n } v = w = { w β := p J un ( β ) | | β | = n } S c = χ (0) ( P ( ~ ) − E ) , S c = χ ( n ) ( P ( ~ ) − E ) (6.2.2)Moreoever | I | = | J | = K n with p J un ( β ) ≤ e nλ max ( d − / . Note that thanks to Proposition4.3.3, c cone ≤ Ch − ( d − cδ long ) / for all ~ sufficiently small;. For our purely logarithmic modes Ψ = Ψ ~ , we can set W = ~ L for L sufficiently large thank to our microlocalization property Lemma 3.2.1. Therefore, for ~ ≤ ~ ( δ long , δ Ehr , K, λ max ), we have established a bound of p (Ψ ~ , ρ , v ) + p (Ψ ~ , τ , w ) ≥ − ( d − cδ long ) | log ~ | − C + O ( ~ L/ ) -LOGARITHMIC MODES 27 where the implicit constant is independent of n and the lower power of L/ V and | I | = K n . 7. Subadditivity statements
Subadditivity of quantum entropies.
Lemma (Approximate shift-invariance for metric entropy on finite sequences) . Fixsome n ≥ , δ Ehr , and ε slab . Given ε < λ max ) and an ε -logarithmic family { Ψ ~ } ~ , thereexists ~ ( n , δ Ehr , ε slab ) such that for all ~ ≤ ~ , any β of length n , and n in the range [0 , T δ Ehr ,ε slab , ~ − n ] we have the estimate H n + n − n ( µ ~ ) = H n − ( µ ~ ) + F entropy ( ~ , ε, n , n ) where F entropy ( ~ , n , n ) := − ε n | log ~ | X | β | = n − log ( k Π β Ψ ~ k ) Im h Π β ( P ( ~ ) − E ) Ψ ~ , Π β Ψ ~ i + 2 ε n | log ~ | X | β | = n Im h Π β ( P ( ~ ) − E ) Ψ ~ , Π β Ψ ~ i + O (cid:18) ε n | log ~ | (cid:19) × X | β | = n k Π β k log (cid:0) k Π β Ψ ~ k (cid:1) + O (cid:18) ε n | log ~ | (cid:19) × X β k Π β k + K ( ~ ) , where K ( ~ ) = O n (cid:16) ε n | log ~ | (cid:17) and the implicit constant depends only on k Π β k .A similar set of formulas hold for H − n − n +1 − n (˜ µ ~ ) with the instances of forward quasipro-jectors replaced by backwards quasiprojectors.Proof. Thanks to the shift-invariance statement Lemma 5.2.2, we have that H n + n − n ( µ ~ ) = − X β ∈A n X α ∈A n k Π β ( n )Π α Ψ ~ k ! log X α ∈A n k Π β ( n )Π α Ψ ~ k ! = − X β ∈A n (cid:0) µ ~ ([ · β ]) + O ( F ( ~ )) (cid:1) log (cid:0) µ ~ ([ · β ]) + O ( F ( ~ )) (cid:1) Next, we use that k Π β Ψ ~ k 6 = 0 (otherwise, we consider only the non-zero elements of thequantum partition, which still makes a probability cylinder measure up to O ( ~ ∞ ) ) terms).Thanks to this, we can expand η ( s ) = − s log s at any point away from 0 to get η (cid:0) k Π β Ψ ~ k + F ( ~ , β , ε, n ) (cid:1) = η (cid:0) k Π β Ψ ~ k (cid:1) + 2 nε | log ~ | (cid:16) log (cid:0) k Π β Ψ ~ k (cid:1) Im h Π β ( P ( ~ ) − E ~ ) Ψ ~ , Π β Ψ ~ i + Im h Π β ( P ( ~ ) − E ~ ) Ψ ~ , Π β Ψ ~ i (cid:17) + O (cid:18) ε n | log ~ | (cid:19) k Π β k log (cid:0) k Π β Ψ ~ k (cid:1) + O (cid:18) ε n | log ~ | (cid:19) k Π β k Summing in β completes our proof. (cid:3) Lemma (Approximate sub-additivity of metric entropy) . Fix some n ≥ , δ Ehr , and ε slab . Given ε < λ max and a ε -logarithmic family Ψ ~ , there exists ~ ( n , δ Ehr , ε slab ) such thatfor all ~ ≤ ~ , any β of length n , and n in the range [0 , T δ Ehr ,ε slab , ~ − n ] we have theestimate H n + n − ( µ ~ ) ≤ H n − ( µ ~ ) + H n − ( µ ~ ) − X | β = n F ( ~ , β , ε, n ) log ( µ ~ ([ · β ]) + F ( ~ , ε, n, n ) where F ( ~ , β , ε, n ) is defined in Lemma 5.2.2 and F ( ~ , ε, n, n ) := O X | α | = n η n X j =1 K j − ! · O ( ~ ∞ ) ! + X | β | = n k Π β Ψ ~ k O (cid:0) log k Π β Ψ ~ k × K n O ( h ∞ ) (cid:1) + X | β | = n F ( ~ , β , n ) O (cid:0) log k Π β Ψ ~ k × K n O ( h ∞ ) (cid:1) . A similar formula holds for H − n − n +1 − n (˜ µ ~ ) in the same vein the previous lemma.Remark . Given our parameter restrictions, F = O n ( h ∞ ). Proof.
First, we keep in mind that P β Π ∗ β ( n )Π β ( n ) = S n + O ( ~ ∞ ) from Proposition 4.1.14which implies X β (cid:0) k Π β ( n )Ψ ~ k + K n × O ( ~ ∞ ) (cid:1) = 1thanks to our microlocalization result Lemma 3.2.1. Therefore, we can write the expression − X | α | = n X | β | = n k Π β ( n )Π α Ψ ~ k log (cid:18) k Π β ( n )Π α Ψ ~ k k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:0) k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:1)(cid:19) which is just H n + n − ( µ ~ ) written out explicitly. Using the homomorphism property oflogarithms and that 1 = k Π β Ψ ~ k + K n ×O ( ~ ∞ ) k Π β Ψ ~ k + K n ×O ( ~ ∞ ) we arrive at − X | α | = n X | β | = n (cid:0) k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:1) k Π β ( n )Π α Ψ ~ k ( k Π β Ψ ~ k + K n × O ( ~ ∞ )) × log (cid:18) k Π β ( n )Π α Ψ ~ k k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:19) − X | α | = n X | β | = n k Π β ( n )Π α Ψ ~ k log (cid:0) k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:1) . -LOGARITHMIC MODES 29 On the first sum, we use that η ( s ) is a convex function and apply Jensen’s Inequality in β , resulting in − X α η X β k Π β ( n )Π α Ψ ~ k ! = − X α η (cid:0) k Π α Ψ ~ k (cid:1) + O X | α | = n η O ( ~ ∞ ) × n X j =1 K j − ! after using the compatibility condition Lemma 5.1.3 via summing in β and then applying aTaylor expansion again. This resulting quantity is approximately H n − ( µ ~ ) + O ( ~ ∞ ).On the second sum, we apply the shift-invariance statement in Lemma 5.2.2 after firstsumming in α , leading us to − X | β | = n (cid:0) k Π β Ψ ~ k + F ( ~ , β , n ) (cid:1) log (cid:0) k Π β Ψ ~ k + K n × O ( ~ ∞ ) (cid:1) = − X | β | = n k Π β Ψ ~ k log k Π β Ψ ~ k − X | β | = n F ( ~ , β , ε, n ) log k Π β Ψ ~ k + X | β | = n F ( ~ , β , ε, n ) × K n × O ( h ∞ ) . This completes our proof. (cid:3)
Subadditivity of quantum pressures.
Lemma (Continuity of potential part of pressure) . Consider the weights w αβ as definedin (6.2.2). Then under the same hypotheses as Lemma 7.1.2, we have that − X α = n X β = n µ ~ ([ · αβ ]) log( w αβ ) = − X α = n µ ~ ([ · α ]) log( w α ) − X β = n µ ~ ([ · β ]) log( w β ) + F ( ~ , ε, n, n ) where F ( ~ , ε, n, n ) := − X β = n w β ) F ( ~ , β , ε, n ) + K n · O ( ~ ∞ ) + n X j =1 K j − ! O ( ~ ∞ ) . The same formula holds for ˜ µ ~ . The proof is very similar to that of the previous lemma and in fact, we have an im-proved compatibility condition: P | β | = n µ ~ ([ · αβ ]) = µ ~ ([ · β ]) + (cid:16)P n j =1 K j − (cid:17) O ( ~ ∞ ) thanksto not having to account for commutators. Next, using Lemmas 5.2.2 and 7.1.1 along withquantities defined in (6.2.2) we easily arrive at the main proposition of Section 7: Proposition [Approximate sub-additivity of quantum pressure functional]Fix some n ≥ , δ Ehr , and ε slab . There exists a real number R > independent of δ Ehr , ε slab and function R ( • , • , • ) on N × (0 , such that ∀ n, n ∈ N with the restriction that n + n ≤ T ε,δ, ~ , lim sup ~ → | R ( n, n , ~ ) | ≤ R along with the following additional properties. Given ε < λ max and an ε -logarithmic family { Ψ ~ } ~ , there exists ~ ( n , δ Ehr , ε slab , P ) such thatfor all ~ ≤ ~ , the pressures with associated weights v, w satisfy p n + n − ( µ ~ , w ) ≤ p n − ( µ ~ , w ) + p n − ( µ ~ , w ) + R ( ~ , ε, n, n ) where R ( ~ , ε, n, n ) = − X | β | = n F ( ~ , β , ε, n ) log ( µ ~ ([ · β ]) + F ( ~ , ε, n, n ) + F ( ~ , ε, n, n ) . The same inequality holds for p n + n (˜ µ ~ , v ) with a corresponding remainder term of ˜ R .Proof. We leave the proof to the interested reader. (cid:3) Pressure lower bounds, Subprincipal functionals, and the final steps
Penultimate bound.
We can rewrite our application of the entropic uncertainty prin-ciple, whose relevant quantities are given in (6.2.2), using our the subadditivity results fromthe previous section as follows:
Proposition (Penultimate pressure bound at finite time n ) . Let µ ~ , ˜ µ ~ be the for-ward/backward quantum symbolic measures associated with ε -logarithmic modes Ψ ~ . Let δ Ehr , ε slab > be the Ehrenfest time parameters. Fix some n > , and for ~ < ~ split the“Ehrenfest” time n = ⌊ T δ Ehr ,ε slab , ~ ⌋ into n = qn + r for r ∈ [0 , n ) .Then the pressures associated with these measures and corresponding choices (6.2.2) satisfythe lower bound: q (cid:0) p n − ( µ ~ , w ) + p n − (˜ µ ~ , v ) (cid:1) + p r − ( µ ~ , w ) + p r − (˜ µ ~ , v ) ≥− ( d − cδ ) × λ + ( ε slab ) × | log ~ | − q − X j =1 R ( ~ , ε, ( q − j ) n , n ) − q − X j =1 ˜ R ( ~ , ε, ( q − j ) n , n ) − R ( ~ , ε, qn , r ) − ˜ R ( ~ , ε, qn , r ) . (8.1.2) Proof.
This is simply an application of Proposition 7.2.2 q times. Two iterations of thisprocess yields p n − ( µ ~ , w ) ≤ p r − ( µ ~ , w ) + p qn − ( µ ~ , w ) + R ( ~ , ε, qn , r ) ≤ p r − ( µ ~ , w ) + p n − ( µ ~ , w ) + p ( q − n − ( µ ~ , w ) + R ( ~ , ε, qn , r ) + R ( ~ , ε, ( q − n , n ) . We leave it to the reader to finish the remaining q − (cid:3) Sub-principal quantities.
Before proceeding any further with calculations, we mustdefine some important (yet technical) quantities that are packaged in the following definition.It is crucial to note that these upcoming functionals, depending on q and n , appear afterdividing the left-hand side of (8.1.2) by q and evaluating the corresponding sums in theremainder terms R, ˜ R . -LOGARITHMIC MODES 31 Definition
Given { Ψ ~ } ~ an ε -logarithmic family, q, n ∈ Z + , and 0 ≤ l ≤ m , we candefine the following quantities: For a ∈ C ∞ ( T ∗ M ), we consider the signed linear functionals m ~ ([ · β ] , ε, q, n , m − l, l ) := q m n m ~ m h| Π β | ( P ( ~ ) − E ~ ) l Ψ ~ , ( E ~ − P ( ~ )) m − l Ψ ~ i and its semiclassical limit m sc ([ · β ] , ε, n , m − l, l ) := lim sup ~ → sup { q : qn ≤⌊ | log ~ | λ max ⌋ } m ~ ([ · β ] , q, n , m − l, l ) . The sub-principal quantum functional is then defined as µ ~ ([ · β ] , ε, q, n ) := − × Im m ~ ([ · β ] , q − , n , m − l, l )+ ∞ X m =2 i m m ! m X l =0 (cid:18) ml (cid:19) m ~ ([ · β ] , q, n , m − l, l ) ! and in parallel, the sub-principal semiclassical functional is µ sc ( E β , n ) := lim sup ~ → sup { q ∈ Z + : qn ≤⌊ | log ~ | λ max ⌋} µ ~ ([ · β ] , ε, q, n ) . With these quantities, we can finally define the sub-principal quantum entropy as H n − ( µ ~ , q ) := − X | β | = n µ ~ ([ · β ] , ε, q, n ) log ( µ ~ ([ · β ]))and the associated ~ -independent quantity H n − ( µ sc ) := X | β | = n − µ sc ( E β , n ) log ( µ sc ([ · β ]))= ∞ X m =1 i m m ! m X l =0 (cid:18) ml (cid:19) X | β | = n m sc ([ · β ] , n , m − l, l ) log ( µ sc ([ · β ])) It is important to emphasize that our various functionals are signed and therefore maynot overlap with measures on cylinder spaces due to the action of q m n m ~ m ( P ( ~ ) − E ~ ) m on anormalized state. Furthemore, this action on Ψ ~ effectively reweights its spectral coefficientstherefore potentially changing its corresponding semiclassical measure. However, there isstill some convenient structure to them and which we explain in the following technicalstatement. First, recall Assumption A described in Section 5.3. Proposition
Let ε < λ max and µ ~ , ˜ µ ~ denote the forward/backward quantum symbolicmeasures associated for some given ε -logarithmic family { Ψ ~ } ~ . Fix some n ≥ , δ Ehr , and ε slab . Let β be of length n . Then (1) We have for all l ≤ m , m sc ([ · β ] , ε, n , m − l, l ) < ∞ , with m sc being a signed, g t -invariant linear functional on C ∞ ( T ∗ M ) with Supp m sc ⊂E . (2) The limiting functional µ sc ([ · β ] , n ) = lim sup ~ → sup { q | qn ≤ | log ~ | λ + } µ ~ ([ · β ] , ε, q, n ) < ∞ with this bound being uniform in n , (3) µ ( • , n ) sc is also a signed g t -invariant linear functional. Under Assumption A , wehave µ sc ( • , n ) << µ sc ( • ) (and therefore Supp µ sc ⊂ E ), and there exists a positivefunction G ( ε ) = 2( e ε − such that µ sc ([ · β ] , n ) ≤ G ( ε ) µ sc ([ · β ]) uniformly in n . Thus, the functional µ sc ( • ) := lim sup n µ sc ( • , n ) exists. (4) And finally, under Assumption A , we have the entropy-type bound of H ( µ sc ) := lim sup n →∞ n lim sup ~ → sup { q : qn ≤⌊ T ⌋} H n − ( µ ~ , q ) !! ≤ G ( ε ) H KS ( µ sc ) . The analogous statements hold for the corresponding functionals for ˜ µ ~ .Proof. We immediately see that m ~ generates a sequence of bounded linear functionals on C ∞ ( T ∗ M ) after using thatsup l ≤ m, { q : qn ≤⌊ | log ~ | λ max ⌋} | m ~ ([ · β ] , q, n , m − l, l ) | ≤ ε m λ m max + o (1) (8.2.3)by the ε -logarithmic property. More specifically, Cauchy-Schwarz gives us an upperbound of q m n m ~ k Π β ( P − E ~ ) m − l Ψ ~ kk Π β ( P − E ~ ) l Ψ ~ k with k ( P − E ~ ) l Ψ ~ k ≤ ε l ~ l | log ~ | l for ~ satisfying the ε -logarithmic property for each l ≤ m .Passing to the Wigner measure for each factor and using that k σ (Π β ) k ∞ ≤ k q l n l ~ l ( P − E ~ ) l Ψ ~ k ≤ ε l andapplying the standard elliptic right-parametrix argument (see the proof of Lemma 3.2.1). For a ∈ C ∞ ( T ∗ M ) supported away from E , we have that k Op ~ ( a )( P ( ~ ) − E ) + iQ ( ~ )) − ( P ( ~ ) − E ~ ) q m − l n m − l ~ m − l ( P − E ~ ) m − l Ψ ~ k ≤ ε m − l +1 ~ .To show m sc ([ · β ] , n , m − l, l ) is g t -invariant, we use the standard idea of commutators asfollows: for q m n m ~ m × i ~ h ( E ~ − P ( ~ )) m − l (cid:2) | Π β | , ( P ( ~ ) − E ) (cid:3) ( P ( ~ ) − E ~ ) l Ψ ~ , Ψ ~ i , the principal symbol of i ~ [ | Π β | , ( P ( ~ ) − E )] being ddt ( π β ◦ g t ) | t =0 = H p π β . Now take the ad-joint of ( E ~ − P ( ~ )) m − l and expand out the commutator. Next, use that { q l n l ~ l ( P − E ~ ) l Ψ ~ } ~ is a family of (non-normalized) quasimodes of width ε ~ | log ~ | with central energies { E ~ } ~ , forall l between 1 and m , to show this limit is 0. Thanks to the absolutely convergent sumdefining µ ~ ( • ), µ sc ( • , n ) is immediately g t -invariant. This completes the proof for (1).By Assumption A , we have that | F ( • , ε ) | << µ sc ( • ) as linear functionals. This state-ment passes immediately onto µ sc ( • , n ) as for β of length n , we have µ ~ ( β , ε, q, n ) = -LOGARITHMIC MODES 33 q P q − j =1 F ( ~ , β , ε, ( q − j ) n ) + o n (1) with an application of the triangle inequality providingthe desired bound; see (8.3.1) for more details. Hence, µ sc ( • , n ) << µ sc ( • ).Thanks to this absolute continuity of measures (with µ sc ( • , n ) being signed), there exists G ( ε ) such that µ ([ · β ] , n ) sc = Z E π β G + dµ sc − Z E π β G − dµ sc and G ± ≤ G ( ε ) by the Hahn-Jordan Decomposition Theorem for linear functionals overcompact, Hausdorff spaces. The reasoning is as follows.We know that each ( G + − G − )( n ) ∈ L ( µ sc ). Thanks to g t being ergodic with respect to µ sc , we see that the invariance of ( G + − G − ) µ sc has two consequences: 1) the invariance ofeach G ± ( n ) on the support of its respective ergodic component of µ sc , and 2) G + − G − is aconstant µ sc -almost everywhere on each of its ergodic components. The bound (8.2.3) turnsinto a bound on these multiplicative factors. Using the estimate (8.2.3) and the definitionof µ ~ ( • ), which when combined provide an absolutely convergent sum, we can take G ( ε ) = 2 ∞ X j =1 j ε j j ! = 2( e ε − . This completes the proof for (2).As the sub-principal quantum functional is convergent, therefore the sub-principal semi-classical functional is well-defined and finite. In turn, we have the double limitlim sup n →∞ n lim sup ~ → sup { q : qn ≤⌊ | log ~ | λ max ⌋} H n − ( µ ~ , q ) = lim sup n →∞ n H n − ( µ sc )= H ( µ sc )being finite and bounded above by G ( ε ) · H KS ( µ sc ). This completes (3) and the remainderof our proof. (cid:3) Remark . Note that we can provide a proof of the absolute continuity of the “compo-nent” functionals m ~ ( • ) in the cases of m = 0 and l = 0 in order to show the spectral weightsdirectly influence the profile of G ± . This can be done in two ways. The first is noticing that m ~ ([ · β ] , q, n , m,
0) = q m n m ~ m h| Π β | ( P − E ~ ) m Ψ ~ , Ψ ~ i = q m n m ~ m h ( P − E ~ ) m Ψ ~ , | Π β | Ψ ~ i and then applying the triangle under the assumption that µ ~ ([ · β ]) = o (1); the second proofis given in the remark below.Some possible ways forward are a generalization of G¨arding’s Inequality to off-diagonalmatrix elements (i.e. a lower bound for h Op ~ ( a ) u, v i ), a more clever application of either thesecond-microlocal calculus described in [35] or the Lagragian foliation -adapated calculus onsurfaces appearing in [15, 6, ? ] to estimate the corresponding phase-space volumes, or a use ofAnantharaman’s robust submultiplicativity argument [1, Section 2.2] classifying dynamicalsequences β . (cid:3) Remark . While it is natural to believe that each of the “component” functionals m sc ( • )is absolutely continuous with respect to µ sc , it appears difficult to establish in all parameters. The main obstruction is showing that for Π β such that µ ~ ([ · β ]) = o (1) in ~ , we havethat sup l ≤ m, { q : qn ≤⌊ T ⌋} | m ~ ([ · β ] , q, n , m − l, l ) | = o (1) as well. This appears to require aknowledge of the asymptotics of µ ~ away from the support of µ sc for which our currentmicrolocal technology does not penetrate. Note that we can provide a proof of the absolutecontinuity in the cases of m = 0 and l = 0 in order to show the spectral weights directlyinfluence the profile of F ± . This can be done in two ways. The first is noticing that m ~ ([ · β ] , q, n , m,
0) = q m n m ~ m h| Π β | ( P − E ~ ) m Ψ ~ , Ψ ~ i = q m n m ~ m h ( P − E ~ ) m Ψ ~ , | Π β | Ψ ~ i and then applying the triangle under the assumption that µ ~ ([ · β ]) = o (1); the second proofis given below.The second proof for the decay of the matrix elements, for m = 0 or l = 0, is moretedious yet uses the spectrum more directly. The corresponding element m when expandedout into eigenfunctions ψ k with spectrum in I ( ε, ~ ), is a sum of further matrix elements c j h| Π β | ψ j , Ψ ~ i each weighted by n ~ ( E j − E ~ ). In other words, we have a linear functionalon the space of energy vectors, that is on R dim Π I . We execute this is only in the case of m = l = 1 with the result for m − l = 0 and l = 0 following similarly.Let E = ( E , . . . , E dim Π ) with the ordering chosen according to that on the real line. Wecan rewrite this leading-order matrix element as ℓ Ψ ~ , β ( E − E ~ ) = X E j ∈ I ( ~ ) ( E j − E ~ ) c j h| Π β | ψ j , Ψ ~ i and whose absolute value is bounded above by k Π β k (cid:13)(cid:13) n ~ ( P − E ~ ) Ψ ~ (cid:13)(cid:13) = O n (cid:16) ε ~ | log ~ | (cid:17) .Notice that µ ~ ([ · β ]) = ℓ Ψ ~ , β ( ). We start with εℓ Ψ ~ , β ( ) − n ~ ℓ Ψ ~ , β ( E − E ) = ℓ Ψ ~ , β ( ε − n ~ ( E − E )) > ε small enough. This follows immediately from the following two steps:1) choose ε = ε × λ + ( ε slab )2(1 − δ Ehr ) so that ε − n ~ ( E j − E ~ ) > ε − ε − δ Ehr ) λ + ( ε slab ) > , which is clearly uniform in n and j thanks to | E j − E ~ | ≤ ε ~ | log ~ | , and 2) observe thatwe know have εℓ Ψ ~ , β ( ) − n ~ ℓ Ψ ~ , β ( E − E ) ≥ ℓ Ψ ~ , β (cid:16)(cid:16) ε − ε − δ Ehr ) λ + ( ε slab ) (cid:17) (cid:17) >
0. Similarly, ℓ Ψ ~ , β ( ε + n ~ ( E − E )) leads to coefficients of the form ε + 2 n ~ ( E j − E ~ ) > ε − ε − δ Ehr ) λ + ( ε slab ) > , also uniform in n and j again thanks to | E j − E ~ | ≤ ε ~ | log ~ | . (cid:3) Corollary
Let µ ~ , ˜ µ ~ denote the forward/backward quantum symbolic measures as-sociated for some given ε -logarithmic family Ψ ~ . Fix some n ≥ , δ Ehr , and ε slab . If ε = o (1) , there exists ~ ( n ) such that for all ~ ≤ ~ ( n ) , any β of length n , and n in therange [0 , T δ Ehr ,ε slab , ~ − n ] we have | µ ~ ([ · β ] , ε, q, n ) | = o (1) and (5.2.3) is a true asymptotic in ~ . Hence the corresponding limiting quantities in Defi-nition 8.2.1 are identically 0. -LOGARITHMIC MODES 35 From F to subprincipal quantities in the lower bound. In this section, we com-pute the remainder sums appearing in Proposition 8.1.1, namely P q − j =1 R ( ~ , ε, ( q − j ) n , n )and P q − j =1 ˜ R ( ~ , ε, ( q − j ) n , n ), and express them in terms of the subprincipal quantitiesdescribed in Definition 8.2.1.We must divide both sides of (8.1.2) by 2 qn in order to compute the Kolmogorov-Sinaientropy at finite times n as well as to compensate for the factor | log ~ | . In fact, it isstraightforward to see that for some ε ′ > q q − X j =1 R ( ~ , ε, ( q − j ) n , n )= − X | β | = n q q − X j =1 F ( ~ , β , ε, ( q − j ) n ) log µ ~ ([ · β ]) − X | β | = n ( µ ~ ([ · β ] , ε, q, n ) + o n (1)) log ( µ ~ ([ · β ]) − X β = n w β ) ( µ ~ ([ · β ] , ε, q, n ) + + o n (1))+ O n ( ~ − ε ′ ) (8.3.1)with a similar expression for q P q − j =1 ˜ R ( ~ , ε, ( q − j ) n , n ); the o (1) error term is ~ . The term O n ( ~ ∞ ) is derived as follows. Note of the formulas for F provided in Section 7 leading to12 qn q − X j =1 F ( ~ , ε, ( q − j ) n , n ) ! = O n ( ~ ∞ ) . For r ≤ n , we see R ( ~ , ε, qn , r ) / qn = ˜ R ( ~ , ε, qn , r ) / qn = O ε | log ~ | × X | β | = r k Π β k log (cid:0) k Π β k (cid:1) . Therefore, as ~ ≤ h , R ( ~ , ε, qn , r ) / qn = O r (cid:16) ε | log ~ | (cid:17) .In totality, we have arrived at Corollary
Let µ ~ , ˜ µ ~ be the forward/backward quantum symbolic measures associ-ated with some ε -logarithmic family { Ψ ~ } ~ . Let δ Ehr , ε slab > be the Ehrenfest time param-eters. Fix some n > , and for ~ < ~ (depending on n amongst other quantities) split the“Ehrenfest” time n = ⌊ T δ Ehr ,ε slab , ~ ⌋ into n = qn + r for r ∈ [0 , n ) . Then the quantum pressures associated with these quantum measures and weights w β = v β = J un ( • ) as in (6.2.2) satisfy the lower bound n (cid:0) p n − ( µ ~ , w ) + p n − (˜ µ ~ , v ) (cid:1) ≥ − ( d − cδ long )2 × λ + ( ε slab ) | log ~ | qn − n (cid:16) H n − ( µ ~ , q ) + H − − n (˜ µ ~ , q ) (cid:17) − n X | β | = n log( J un ( β )) × µ ~ ([ · β ] , ε, q, n ) + X | β | = n log( J un ( β )) × ˜ µ ~ ([ · β ] , ε, q, n ) + O r (cid:18) ε | log ~ | (cid:19) + O n ( ~ − ε ′ ) + o ( n − ) + o (1) . (8.3.3)8.4. Parameter selections and reductions to classical pressure.
We now execute thefinal steps en route to completing the proof of Theorem 1.2.1. Recall the notion of entropyfor smooth partitions as discussed in Section 4.1.3. While we proceed exactly as in [3, Section2.2.8] and [29, Section 3.10], we present only the main details for readability. First recallDefinition 8.2.1 on our various subprincipal quantities.Let qn = ⌊ T δ Ehr ,ε slab , ~ ⌋ − r = O r ( ε slab ,δ Ehr ) λ max | log ~ | . Set ε slab = δ Ehr (and recall that ε sc = ε slab ) and let δ long < δ Ehr . Furthermore, let the diameter of our initial dynamicalsmooth partition P (the same given in [1, Appendix A2] and whose boundaries are notcharged by µ sc ) be a small parameter ζ and note that K ≈ ζ − d . Finally, ε > δ Ehr , ζ small enough, n large enough, and finally a small enoughnumber ~ ( δ Ehr , ζ , n , ε ) such that for all ~ ≤ h the terms depending on ~ in (8.3.3) arewithin ε of their respective limits in ~ , that is1 n H ( µ sc , P ∨ n sm ) − n X | β | = n log( J un ( β )) × µ sc ( g n ( π β )) + X | β | = n log( J un ( β )) × µ sc ( π β ) ≥ − ( d − λ max − n (cid:16) H n − ( µ sc ) + H − − n (˜ µ sc ) (cid:17) − n X | β | = n log( J un ( β )) × µ sc ([ · β ] , n )) − n X | β | = n log( J un ( β )) × ˜ µ sc ([ · β ] , n ) + o ( n − ) + O ( δ Ehr ) + O ( ε ) . Above, we have invoked the g t -invariance of our functionals of µ sc , recalled Lemma 7.2.1 totrack the factors of 2, and recalled the formulas for w β to track the powers of 1 /
2. Theproperty that our initial partition P does not charge the boundaries of the elements E k anda convolution argument involving P sm allows us to replace all instances of smooth cutoffs π β above with the sharp indicator functions on E β . At this point, we remind ourselves of keypoints: − X | β | = n log( J un ( β )) × µ sc ( E β ) = − n X E k ∈P µ sc ( E k ) log( J u ( k )) -LOGARITHMIC MODES 37 thanks to the multiplicative structure of J un as described in (4.3.2) and that µ sc is g t -invariantmeasure. As µ sc is also g t -invariant signed measure by Lemma 8.2.2, we similarly have thatfor each n − X | β | = n log( J un ( β )) × µ sc ( E β , n ) = − n X E k ∈P µ sc ( E k , n ) log( J u ( k ))with corresponding expressions for the backwards measures. Thanks to our parameterchoices, there exists a subsequence in the tuple of parameters ( δ Ehr , ζ , n , ~ ) such thatTheorem 1.2.1 holds for the sharp indicator functions. Now we replace the “fattened” µ sc given by C d r d − ε sm dr ⊗ dµ ( ρ ) and defined on specific neighborhood of E ε sm , with thetrue measure supported on E via the upper semi-continuity of H KS . Now, recall that µ sc ( • ) := lim sup n µ sc ( • , n ).It is tedious, but not hard, to show that H ( µ sc ) (as given in Propostion 8.2.2) is upper semi-continuous in µ sc as well. It is crucial that our fattened µ sc converges in total variation to µ sc .To be more precise, let µ sc,ε sm be the described sequence of fattened measures converging to µ sc . The main estimate is on the differences m sc,ε sm ( • ) log µ sc,j ( • ) − m sc ( • ) log µ sc ( • ) . Thanksto the explicit formulas given in Definition 8.2.1, it is straightforward to show there exists afunction g continuous near 0 such that | m sc ( • ) − m sc,ε sm ( • ) | ≤ g ( ε sm ) | min { m sc ( • ) , m sc,ε sm ( • ) }| for ε sm sufficiently small. Next, we use that lim sup n n H n − ( µ sc ) ≤ G ( ε ) H KS ( µ sc ) to showthat lim sup n − n X | β | = n ( m sc,ε sm ( • ) log µ sc,j ( • ) − m sc ( • ) log µ sc ( • )) ≥ P ∞ m =1 P ml =0 (cid:0) ml (cid:1) of the above limit in n to recover µ sc ( • )and then take the limit supremumin ε sm . This completes the proof of our main result Theorem 1.2.1.The proof of Corollary 1.2.2 follows simply from the inequalities | Z E log J u ( ρ ) d µ sc ( ρ ) | ≤ G ( ε ) Z E log J u ( ρ ) d µ sc ( ρ )and H ( µ sc ) ≤ G ( ε ) H KS ( µ sc ) , themselves following from our Proposition 8.2.2. Now apply the Ruelle upperbound and thatmax µ ∈ M R E log J u ( ρ ) d µ ( ρ ) = d − M is the space of g t -invariant measures on E . Acknowledgements:
SE would like to thank Zeev Rudnick, Filippo Morabito, GabrielRivi`ere, and Lior Silberman for comments. In particular, SE warmly thanks St´ephane Non-nenmacher for various discussions during the writing of this paper.
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