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On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

Gabriel Rivière, Lasse L. Wolf

TL;DR

The article addresses semiclassical limits for joint eigenfunctions of abelian quantum actions quantizing symplectic toral automorphisms. It introduces a robust framework combining metaplectic torus quantization, entropy bounds, and rigidity results to show that, in the irreducible case with a suitable eigenvalue condition, semiclassical measures decompose as a convex sum of Lebesgue and a zero-entropy component with Lebesgue weight at least 1/2; in the reducible case, Lebesgue components along invariant subtori carry at least half the mass. The work leverages Dirichlet-type structure results for centralizers, Einsiedler-Lindenstrauss rigidity, and entropy methods to obtain quantitative lower bounds on Lebesgue components and to describe admissible decompositions in terms of invariant subspaces. It also situates these findings relative to Hecke operator frameworks and provides conditions (Galois-type) and explicit constructions to realize the hypotheses, thus connecting quantum chaos on tori with arithmetic dynamics and rigidity phenomena.

Abstract

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$.

On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

TL;DR

The article addresses semiclassical limits for joint eigenfunctions of abelian quantum actions quantizing symplectic toral automorphisms. It introduces a robust framework combining metaplectic torus quantization, entropy bounds, and rigidity results to show that, in the irreducible case with a suitable eigenvalue condition, semiclassical measures decompose as a convex sum of Lebesgue and a zero-entropy component with Lebesgue weight at least 1/2; in the reducible case, Lebesgue components along invariant subtori carry at least half the mass. The work leverages Dirichlet-type structure results for centralizers, Einsiedler-Lindenstrauss rigidity, and entropy methods to obtain quantitative lower bounds on Lebesgue components and to describe admissible decompositions in terms of invariant subspaces. It also situates these findings relative to Hecke operator frameworks and provides conditions (Galois-type) and explicit constructions to realize the hypotheses, thus connecting quantum chaos on tori with arithmetic dynamics and rigidity phenomena.

Abstract

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic -action with . We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the -action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight .

Paper Structure

This paper contains 25 sections, 31 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Quantum maps
  3. Main results
  4. Comparison with Hecke operators for quantum maps
  5. Organization of the article
  6. Semiclassical measures for abelian actions
  7. Quantum mechanics on the torus
  8. Semiclassical measures
  9. A reminder on ergodic decomposition and Kolmogorov-Sinai entropy
  10. Centralizers of symplectic matrices
  11. Preliminary conventions
  12. The irreducible general linear case
  13. The irreducible symplectic case
  14. The general symplectic case
  15. Reducibility and rigidity of actions on tori
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $d\geq 2$ and $A\in \operatorname{Sp}(2d,\mathbb{Z})$ with irreducible characteristic polynomial over $\mathbb{Q}$ such that no ratio of eigenvalues $\frac{\lambda_i}{\lambda_j}$, $i\neq j$, is a root of unity. Furthermore assume $m(A)+l(A)\geq 2$. Then, for every $\varepsilon >0$, one can find and such that, for any sequence $(\psi_k)_{k\geq 1}$ satisfying one has that, for every $\mu\in\ma

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Remark
  • Remark
  • Theorem 2.1: Riviere11
  • Remark
  • Remark
  • Corollary 2.2
  • Definition 3.1
  • Remark
  • ...and 65 more