On Ruelle's Lemma and Ruelle Zeta Functions

TL;DR

This paper provides a clearer proof of an important inequality related to the Ruelle operator in hyperbolic flows, extending it to higher dimensions and aiding the understanding of Ruelle zeta functions' analyticity.

Contribution

It offers a detailed, generalized proof of an inequality for the Ruelle operator, crucial for analyzing Ruelle zeta functions in higher-dimensional hyperbolic flows.

Findings

Established a generalized inequality for the Ruelle operator in higher dimensions

Clarified the proof process with detailed explanations

Supported the analyticity of Ruelle zeta functions

Abstract

In this article we prove an important inequality regarding the Ruelle operator in hyperbolic flows. This was already proven briefly by Mark Pollicott and Richard Sharp in a low dimensional case, but we present a clearer proof of the inequality, filling in gaps and explaining the ideas in more detail, and extend the inequality to higher dimensional flows. This inequality is necessary to prove a proposition about the analyticity of Ruelle zeta functions.