Hecke theory and equidistribution for the quantization of linear maps of the torus

Abstract

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 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.

The full delocalization of eigenstates for the quantized cat map

“…For instance for γ DE , for every N ∈ 2N + 1, up to some scalar phase factor (its expression as a quotient of Gauss sums is expanded in [KR00] and [Mez02]),…”

Matrix elements for the quantum cat map: fluctuations in short windows

“…The only rigorous results available concern special arithmetic systems, namely cat maps and some special compact surfaces of constant negative curvature, uniformized by unit groups of rational quaternion algebras. In these cases many quantum symmetries exist, and QUE is now known to hold for eigenfunctions of the desymmetrized system [17,23]. The complexity of the problem increases as we increase the number of degrees of freedom and Kelmer [15] found systematic deviations from QUE for higher dimensional cat maps.…”

Macroscopic limits of chaotic eigenfunctions

“…There is also an analogue of arithmetic Quantum Unique Ergodicity in the setting of cat maps: Kurlberg-Rudnick [KR00] introduced Hecke operators which commute with B N and showed that any sequence of joint eigenfunctions of B N and these operators converges to the Lebesgue measure. This does not contradict the counterexample of Theorem 6 since for the values of N j chosen there, the map B N j has eigenvalues of high multiplicity.…”