Kurlberg P., Rudnick Z.
Eigenfunctions of the Quantized Cat Map
(87K, LATeX 2e)
ABSTRACT. We study semi-classical limits of eigenfunctions of a quantized
linear hyperbolic automorphism of the torus (``cat map'').
For some values of Planck's constant,
the spectrum of the quantized map has large degeneracies.
Our first goal in this paper is to show that these degeneracies are
coupled to the existence of {\em quantum symmetries}. There is a
commutative group of unitary operators on the state-space which commute with
the quantized map and therefore act on its eigenspaces.
We call these ``Hecke operators'', in analogy with the setting of
the modular surface.
We call the eigenstates of both the quantized map and of all the Hecke
operators ``Hecke eigenfunctions''.
Our second goal is to study the semiclassical limit of
the Hecke eigenfunctions. We will show that they become
equidistributed with respect to Liouville measure, that is the
expectation values of quantum observables in these eigenstates
converge to the classical phase-space average of the observable.