Semiclassical measures for higher dimensional quantum cat maps

To explain quantum cat maps in more detail, we use a semiclassical quantization procedure, mapping each classical observable (a function on a symplectic manifold called the phase space) to a quantum observable (an operator on some Hilbert space). In our setting the phase space is the torus $\mathbb{T}^{2n}$ and each classical observable is quantized to a family of operators (see §2.2.3):

Here $\theta\in\mathbb{T}^{2n}$ is a parameter, $\mathbf{N}\geq 1$ is an integer, and the Hilbert spaces of quantum states $\mathcal{H}_{\mathbf{N}}(\theta)$, defined in §2.2.1, have dimension $\mathbf{N}^{n}$. We denote the inner product on these spaces by $\langle\bullet,\bullet\rangle_{\mathcal{H}}$. The semiclassical parameter is $h:=(2\pi\mathbf{N})^{-1}$.

Every matrix $A\in\operatorname{Sp}(2n,\mathbb{Z})$ defines a symplectic automorphism of the torus $\mathbb{T}^{2n}$. This automorphism is quantized by a family of unitary maps

which satisfy the following exact Egorov’s theorem (2.58) intertwining the action of $A$ on $\mathbb{T}^{2n}$ with conjugation by $M_{\mathbf{N},\theta}$:

To construct such $M_{\mathbf{N},\theta}$ we need to impose a quantization condition (2.57) on $\theta$. The constructed operators are unique up to multiplication by a unit complex constant. See §2.2.4 for more details and §2.2.5 for explicit formulas for $\operatorname{Op}_{\mathbf{N},\theta}(a)$ and $M_{\mathbf{N},\theta}$.

Throughout the paper we fix $A\in\operatorname{Sp}(2n,\mathbb{Z})$ and assume the following spectral gap condition on the spectrum $\operatorname{Spec}(A)$:

This condition is crucial in the proof because it means there is a one-dimensional ‘fast’ direction in which the powers of $A$ grow faster than in other directions, which makes it possible to apply the one-dimensional fractal uncertainty principle – see §1.4.

Our first result is the following analog of (1.1):

Here the norm $\|\operatorname{Op}_{\mathbf{N},\theta}(a)u\|_{\mathcal{H}}$ on the right-hand side of (1.4) plays a similar role to the norm $\|u\|_{L^{2}(\Omega)}$ on the right-hand side of (1.1): if $a$ is supported in some $\mathbf{N}$-independent subset of $\mathbb{T}^{2n}$, then $\|\operatorname{Op}_{\mathbf{N},\theta}(a)u\|_{\mathcal{H}}$ can be thought of as the norm of $u$ localized in the position-frequency space to this set.

We remark that the conditions of Theorem 1 are always satisfied if $n=1$ (i.e. $A$ is a $2\times 2$ matrix) and $A$ is hyperbolic, that is it has no eigenvalues on the unit circle. Thus Theorem 1 (or more precisely, Theorem 2 below) implies the result of [Sch21]. See Figure 1 for a numerical illustration in the case $n=1$. For $n\geq 2$, our assumption (1.3) does not require $A$ to be hyperbolic. We also remark that having characteristic polynomial irreducible over $\mathbb{Q}$ is a generic property for integer symplectic matrices, see Rivin [Riv08] and the book of Kowalski [Kow08, Theorem 7.12].

Theorem 1 is a consequence of a more general result, Theorem 2 below, that applies to quasimodes of $M_{\mathbf{N},\theta}$ and does not require the irreducibility over $\mathbb{Q}$ of the characteristic polynomial of $A$. Before stating it, we need to introduce more notation. In order to measure the strength of a quasimode, we introduce for $u\in\mathcal{H}_{\mathbf{N}}(\theta)$ the quantity

where $\mathbb{S}^{1}$ denotes the unit circle in $\mathbb{C}$. Note that $\mathbf{r}_{M}(u)=0$ if and only if $u$ is an eigenfunction of $M_{\mathbf{N},\theta}$.

To relax the assumption on the characteristic polynomial of $A$, let us notice that by (1.3) the leading eigenvalue $\lambda_{+}$ is real. Since the matrix $A$ is symplectic, its transpose is conjugate to its inverse, thus $\lambda_{-}:=\lambda_{+}^{-1}$ is also a simple eigenvalue for $A$ and $|\lambda_{-}|<1<|\lambda_{+}|$. Moreover, all other eigenvalues $\lambda$ of $A$ satisfy $|\lambda_{-}|<|\lambda|<|\lambda_{+}|$.

Denote by $E_{\pm}\subset\mathbb{R}^{2n}$ the (real) eigenspaces of $A$ associated to $\lambda_{\pm}$. Let $V_{\pm}$ be the smallest subspace of $\mathbb{Q}^{2n}$ such that $E_{\pm}$ is contained in $V_{\pm}\otimes\mathbb{R}$. Note that $V_{\pm}$ are invariant under $A$. Denote by

the subtori given by the projections of $V_{\pm}\otimes\mathbb{R}$ to $\mathbb{T}^{2n}$.

The tori $\mathbb{T}_{\pm}$ are relevant here because of their alternative dynamical definition given in Lemma 4.3: if $e_{\pm}$ is any eigenvector of $A$ associated to the eigenvalue $\lambda_{\pm}$, then the closure of the orbit of a point $x\in\mathbb{T}^{2n}$ by the translation flow generated by $e_{\pm}$ is $x+\mathbb{T}_{\pm}$. In our setting, these translation flows will play the role that was played by the horocyclic flows on the unit tangent bundle of hyperbolic surfaces in [DJ18]. Let us also give

We now state a more general version of Theorem 1:

Theorem 2 implies Theorem 1. Indeed, if the characteristic polynomial of $A$ is irreducible over the rationals, then $\mathbb{T}_{\pm}=\mathbb{T}^{2n}$ (see Lemma A.3), thus every nonempty open set satisfies the geometric control condition transversally to $\mathbb{T}_{+}$ and $\mathbb{T}_{-}$. However, one can find a matrix $A$ that satisfies (1.3) but for which (1.4) fails for certain choices of $a$ (in particular the characteristic polynomial of $A$ is reducible over $\mathbb{Q}$) – see the examples in §A.2.

We will also prove the following theorem about damped quantum cat maps. It is analogous to exponential energy decay for negatively curved surfaces proved in [DJN22] (following earlier work of Jin [Jin20] in the constant curvature case):

Note that by Lemma A.3, if the characteristic polynomial of $A$ is irreducible over $\mathbb{Q}$ then the condition on $b$ simplifies to $\{|b|<1\}\neq\emptyset$. See Figure 2 for a numerical illustration.

We now give an application of Theorem 2 to semiclassical measures. These measures describe the possible ways in which the mass of eigenfunctions of $M_{\mathbf{N},\theta}$ can distribute in the position-frequency space in the limit $\mathbf{N}\to\infty$, and they are defined as follows:

One can show (similarly to [Zwo12, Theorem 5.2] using a diagonal argument, the norm bound (2.51), Riesz representation theorem for the dual to $C^{0}(\mathbb{T}^{2n})$, and the sharp Gårding inequality (2.48)) that each norm 1 sequence $u_{j}\in\mathcal{H}_{\mathbf{N}_{j}}(\theta_{j})$ has a subsequence which has a weak limit in the sense of (1.8). Every semiclassical measure is a probability measure on $\mathbb{T}^{2n}$ (as follows from taking $a=1$ in (1.8) and using that $\operatorname{Op}_{\mathbf{N},\theta}(1)=I$) which is invariant under the map $A$ (as follows from (1.2)). Semiclassical measures for quantum cat maps are analogous to those for Laplacian eigenfunctions on a Riemannian manifold $(\mathcal{M},g)$, which are probability measures on the cosphere bundle $S^{*}\mathcal{M}$ invariant under the geodesic flow – see [Zwo12, Chapter 5] for more information.

As explained in §3.3, Theorem 2 implies the following property of the support of semiclassical measures:

In Appendix A, we discuss the algebraic properties of the spaces $V_{+}$ and $V_{-}$, and their implications in terms of the supports of semiclassical measures for $A$. In particular, we give examples of the following situations:

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  the characteristic polynomial of $A$ is irreducible, and hence the semiclassical measures are fully supported;

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  $\mathbb{Q}^{2n}=V_{+}\oplus V_{-}$, in which case there are semiclassical measures supported on translates of $\mathbb{T}_{+}$ and $\mathbb{T}_{-}$ due to [Kel10];

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  $V_{+}+V_{-}\neq\mathbb{Q}^{2n}$ and there are semiclassical measures supported in $\mathbb{T}_{+}$ and $\mathbb{T}_{-}$;

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  $V_{+}+V_{-}\neq\mathbb{Q}^{2n}$ but the supports of semiclassical measures (associated to a certain basis of eigenfunctions) are strictly larger than $\mathbb{T}_{+}$ or $\mathbb{T}_{-}$.

The last two cases emphasize the fact that when $V_{+}+V_{-}\neq\mathbb{Q}^{2n}$, one has to study the action of $A$ on $\mathbb{Q}^{2n}/(V_{+}+V_{-})$ in order to refine the information on the support of semiclassical measures given by Theorem 4.

We now give a brief overview of history of semiclassical measures in quantum chaos, referring the reader to review articles by Marklof [Mar06], Sarnak [Sar11], Nonnenmacher [Non13], and Dyatlov [Dya21] for more information. In the setting of eigenfunctions of the Laplacian on compact negatively curved Riemannian manifolds, the Quantum Ergodicity theorem of Shnirelman [Shn74], Zelditch [Zel87], and Colin de Verdière [CdV85] states that a density 1 sequence of eigenfunctions equidistributes, i.e. converges to the Liouville measure in a way analogous to (1.8). The Quantum Unique Ergodicity conjecture of Rudnick–Sarnak [RS94] claims that the whole sequence of eigenfunctions equidistributes, i.e. the Liouville measure is the only semiclassical measure. It is open in general but known in the particular cases of arithmetic hyperbolic surfaces for joint eigenfunctions of the Laplacian and the Hecke operators, which are additional symmetries present in the arithmetic case – see Lindenstrauss [Lin06] and Brooks–Lindenstrauss [BL14].

The works of Anantharaman [Ana08], Anantharaman–Nonnenmacher [AN07b], Anantharaman–Koch–Nonnenmacher [AKN09], Rivière [Riv10b, Riv10a], and Anantharaman–Silberman [AS13] establish entropy bounds, which are positive lower bounds on the Kolmogorov–Sinaĭ entropy of semiclassical measures. As mentioned in the beginning of the introduction, in the setting of negatively curved surfaces the paper [DJN22] gives another restriction: all semiclassical measures have full support. In the case of hyperbolic surfaces this was previously proved by Dyatlov–Jin [DJ18].

The study of quantum cat maps, which is the setting of the present paper, goes back to the work of Hannay–Berry [HB80]. Bouzouina–De Bièvre [BDB96] proved the analogue of quantum ergodicity in this setting. Faure–Nonnenmacher–De Bièvre [FNDB03] constructed examples of semiclassical measures for 2-dimensional quantum cat maps which are not the Liouville measure; this contradicts the Quantum Unique Ergodicity conjecture for quantum cat maps but it does not contradict Theorem 1 since these measures were supported on the entire $\mathbb{T}^{2}$. Faure–Nonnenmacher [FN04], Brooks [Bro10], and Rivière [Riv11] established ‘entropy-like’ bounds on semiclassical measures; see also Anantharaman–Nonnenmacher [AN07a] and Gutkin [Gut10] for entropy bounds for other models of quantum maps. As mentioned above, Schwartz [Sch21] obtained Theorem 1 for 2-dimensional quantum cat maps.

Kurlberg–Rudnick [KR00] introduced the analogue of Hecke operators in the setting of 2-dimensional quantum cat maps. For the joint eigenfunctions of a quantum cat map and the Hecke operators (known as arithmetic eigenfunctions) they showed that Quantum Unique Ergodicity holds; see also Gurevich–Hadani [GH11]. This does not contradict the counterexample in [FNDB03] since the quantum cat maps used there have eigenvalues of high multiplicity.

In the setting of higher dimensional quantum cat maps which have an isotropic invariant rational subspace, Kelmer [Kel10] constructed examples of semiclassical measures (for arithmetic eigenfunctions) which are supported on proper submanifolds of $\mathbb{T}^{2n}$. On the other hand, if there are no isotropic invariant rational subspaces, then [Kel10] gives Quantum Unique Ergodicity for arithmetic eigenfunctions. Compared to Theorem 4, the conclusion of [Kel10] is stronger than $\operatorname{supp}\mu=\mathbb{T}^{2n}$ and the assumption is weaker than the characteristic polynomial of $A$ being irreducible over rationals (since [Kel10] only assumes that there are no isotropic invariant rational subspaces), however the result of [Kel10] only applies to arithmetic eigenfunctions. For a further discussion of the relation of our results with those of [Kel10], see Appendix A.

We now give an outline of the proof of Theorem 2. For simplicity we assume that the symbol $a$ satisfies $0\leq a\leq 1$, as well as the following stronger version of the geometric control condition: there exists an open set $\mathcal{U}\subset\mathbb{T}^{2n}$ such that $a=1$ on $\mathcal{U}$ and each shifted torus $x+\mathbb{T}_{+}$, $x+\mathbb{T}_{-}$ intersects $\mathcal{U}$. We also assume that $\mathbf{N}$ is large and $u\in\mathcal{H}_{\mathbf{N}}(\theta)$ is an eigenfunction of $M_{\mathbf{N},\theta}$.

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  In §2 we review semiclassical quantization and metaplectic operators on $\mathbb{R}^{2n}$ and on the torus $\mathbb{T}^{2n}$, as well as the anisotropic symbol classes $S_{L,\rho,\rho^{\prime}}$.

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  In §3 we prove Theorems 2–4 modulo the key estimate given by Proposition 3.10.

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  In §4 we prove Proposition 3.10 using the fractal uncertainty principle.

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  Finally, in Appendix A, we discuss the algebraic properties of the spaces $V_{+}$ and $V_{-}$ and investigate the sharpness of Theorem 4 in different situations.