A Ruelle-Perron-Frobenius theorem for expanding circle maps with an indifferent fixed point

TL;DR

This paper extends the Ruelle-Perron-Frobenius theorem to expanding circle maps with an indifferent fixed point, establishing existence and uniqueness of Gibbs measures for certain observables.

Contribution

It introduces a novel approach to analyze transfer operators near indifferent fixed points, proving a unique Gibbs equilibrium state for these systems.

Findings

Existence of a positive eigenfunction associated with the maximal eigenvalue.

Identification of a suitable linear space for transfer operator analysis.

Proof of the uniqueness of the Gibbs measure for the system.

Abstract

In this note, we establish an original result for the thermodynamic formalism in the context of expanding circle transformations with an indifferent fixed point. For an observable whose continuity modulus is linked to the dynamics near such a fixed point, by identifying an appropriate linear space to evaluate the action of the transfer operator, we show that there is a strictly positive eigenfunction associated with the maximal eigenvalue given as the exponential of the topological pressure. Taking into account also the corresponding eigenmeasure, the invariant probability thus obtained is proved to be the unique Gibbs-equilibrium state of the system.