The Analyticity of a Generalized Ruelle's Operator

TL;DR

This paper introduces a generalized Ruelle operator for one-dimensional lattices, extending classical concepts to more complex symbolic systems, and proves its analyticity under broad conditions.

Contribution

It defines a new generalized Ruelle operator that encompasses previous operators and proves its analyticity, broadening the scope of thermodynamic formalism.

Findings

Proves the analyticity of the generalized Ruelle operator.

Extends the concept of transition matrices to variable symbol sets.

Provides examples illustrating the generalized operator's properties.

Abstract

In this work we propose a generalization of the concept of Ruelle operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle operator, that generalizes both the Ruelle operator proposed in [BCLMS] and the Perron Frobenius operator defined in [Bowen]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle operator and present some examples.