MPHA Seminar, Blocker 112, 19th October 2007, 3pm

Speaker: Boris Gutkin, Erlangen-Nuremberg, Germany
Title: Metric bounds on the semiclassical measures of quantized 1d 
maps.

Quantum ergodicity asserts that almost all infinite sequences of
eigenstates of a quantized chaotic system are equidistributed in the
phase space.  On the other hand, there are might exist exceptional
sequences which converge to different (non-Liouville) classically
invariant measures.  Recently, a considerable progress in
understanding of what kind of exceptional semiclassical measures might
appear in quantum systems has been achieved.  By a remarkable result
of N. Anantharaman and S. Nonnenmacher (math-ph/0610019,
arXiv:0704.1564 with H. Koch), the metric entropy H_{KS}(\mu) of any
semiclassical measure \mu arising in a chaotic Hamiltonian system must
be bounded from below by a certain explicitely given constant C.  This
result turns out to be optimal for uniformly expanding systems.
Unfortunately, this is not the case for the systems with varying
expansion rate, where the constant C could even become negative (thus
rendering the bound trivial).  A stronger bound, valid for all
Hamiltonian systems, has been conjectured by N. Anantharaman and
S. Nonnenmacher in the same paper, but so far has not been proven.

In the present work we consider this question using quantized
one-dimensional maps as a model.  For a certain class of non-uniformly
expanding maps we prove Anantharaman-Nonnenmacher conjecture.
Furthermore, for these maps we are able to construct explicit
sequences of eigenstates which saturate the bound.  This demonstrates
that the bound is actually optimal.