On the distribution of zeros of a Ruelle zeta-function

TL;DR

This paper investigates how the eigenvalues of the Ruelle operator, associated with the quadratic map z ↦ z^2 + c, are distributed as c approaches -2 from below, revealing insights into the spectral properties of this dynamical system.

Contribution

It provides a detailed analysis of the limit distribution of eigenvalues of the Ruelle operator for quadratic maps near a critical parameter value.

Findings

Eigenvalues exhibit a specific limiting distribution as c approaches -2.

The spectral behavior of the Ruelle operator is characterized in this parameter regime.

Results contribute to understanding the spectral theory of dynamical systems with quadratic maps.

Abstract

We study the limit distribution of eigenvalues of a Ruelle operator (which is also called the Thurston pushforward operator) for the dynamical system $z \mapsto z^2+c$ when $c<-2$ and tends to $-2$.