Entropic bounds on semiclassical measures for quantized one-dimensional maps

TL;DR

This paper proves a conjectured entropic bound for semiclassical measures in quantized one-dimensional maps, demonstrating its optimality by constructing eigenstates that saturate the bound, thus extending quantum ergodicity results.

Contribution

It establishes the Anantharaman-Nonnenmacher conjecture for certain non-uniformly expanding maps and constructs eigenstates that achieve the bound.

Findings

Proved the entropic bound for specific non-uniformly expanding maps.

Constructed eigenstates that saturate the entropic bound.

Extended quantum ergodicity results to a new class of maps.

Abstract

Quantum ergodicity asserts that almost all infinite sequences of eigenstates of a quantized ergodic system are equidistributed in the phase space. On the other hand, there are might exist exceptional sequences which converge to different (non-Liouville) classical invariant measures. By the remarkable result of N. Anantharaman and S. Nonnenmacher math-ph/0610019, arXiv:0704.1564 (with H. Koch), for Anosov geodesic flows the metric entropy of any semiclassical measure must be bounded from below. The result seems to be optimal for uniformly expanding systems, but not in general case, where it might become even trivial if the curvature of the Riemannian manifold is strongly non-uniform. It has been conjectured by the same authors, that in fact, a stronger bound (valid in general case) should hold. In the present work we consider such entropic bounds using the model of quantized one-dimensional maps. For a certain class of non-uniformly expanding maps we prove Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.