Summary: ENTROPY OF SEMICLASSICAL MEASURES OF THE
WALSH-QUANTIZED BAKER'S MAP
NALINI ANANTHARAMAN AND STÉPHANE NONNENMACHER
Abstract. We study the baker's map and its Walsh quantization, as a toy model of a
quantized chaotic system. We focus on localization properties of eigenstates, in the semi-
classical régime. Simple counterexamples show that quantum unique ergodicity fails for
this model. We obtain, however, lower bounds on the entropies associated with semiclas-
sical measures, as well as on the Wehrl entropies of eigenstates. The central tool of the
proofs is an "entropic uncertainty principle".
In the semiclassical (highly-oscillatory) framework, one can generally express the solu-
tion of the time-dependent Schrödinger equation as an -expansion based on the classical
motion. Classical mechanics is then the 0-th order approximation to wave mechanics.
However, such expansions are not uniform in time, and generally fail to capture the
infinite-time evolution of the quantum system, or its stationary properties. Unless the
system is completely integrable, the instabilities of the classical dynamics will ruin the
semiclassical expansion beyond the Ehrenfest time, which is of order | log |.
Nevertheless, the domain dubbed as "quantum chaos" expresses the belief that strongly
chaotic properties of the classical system induce certain typical patterns in the stationary
properties of the quantum system, like the statistical properties of the eigenvalues (the