Weyl law for contractive maps

TL;DR

This paper establishes a Weyl law for eigenvalues of contractive maps, highlighting its independence from the dimension of the invariant set and proposing non-orthogonality of eigenvectors as a key factor.

Contribution

It introduces a Weyl law for contractive maps and suggests a new explanation based on eigenvector non-orthogonality, differing from classical phase space arguments.

Findings

Eigenvalues follow a specific Weyl law

The law is insensitive to the dimension of the attractor

Eigenvector non-orthogonality explains the spectral behavior

Abstract

We find the Weyl law followed by the eigenvalues of contractive maps. An important property is that it is mainly insensitive to the dimension of the corresponding invariant classical set, the strange attractor. The usual explanation for the fractal Weyl law emergence in scattering systems (i.e., having a projective opening) is based on classical phase space distributions evolved up to the quantum to classical correspondence (Ehrenfest) time. In the contractive case this reasoning fails to describe it. Instead, we conjecture that the support for this behavior is essentially given by the strong non-orthogonality of the eigenvectors of the contractive superoperator.