Compactness of Transfer Operators and Spectral Representation of Ruelle Zeta Functions for Super-continuous Functions

TL;DR

This paper develops a framework for analyzing transfer operators and Ruelle zeta functions for super-continuous functions on topological Markov shifts, extending classical results to a broader class of functions.

Contribution

It constructs a Banach space where transfer operators are compact and derives trace formulas and spectral representations for super-continuous functions, generalizing classical results.

Findings

Established trace formula for super-continuous functions

Derived spectral representation of Ruelle zeta functions

Extended classical results to broader function classes

Abstract

Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.