Ergodic Transport Theory and Piecewise Analytic Subactions for Analytic Dynamics

TL;DR

This paper develops a theory connecting ergodic transport, complex analytic extensions, and subactions for real expanding maps, providing new insights into the structure of eigenfunctions and their limits in dynamical systems.

Contribution

It introduces a framework for analyzing piecewise analytic subactions and eigenfunctions in complex extensions of expanding maps, linking ergodic transport with complex dynamics.

Findings

Existence of real analytic eigenfunctions for the Ruelle operator in complex extensions.

Convergence of scaled logarithms of eigenfunctions to piecewise analytic subactions.

Application to the case where the potential is related to the derivative of the map.

Abstract

We consider a piecewise analytic real expanding map $f: [0,1]\to [0,1]$ of degree $d$ which preserves orientation, and a real analytic positive potential $g: [0,1] \to \mathbb{R}$. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume $\log g$ is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential $\beta \,\log g$, where $\beta$ is a real constant, there exists a real analytic eigenfunction $\phi_\beta$ defined on $[0,1]$ (with a complex analytic extension) for the Ruelle operator of $\beta \,\log g$. Under some assumptions we show that $\frac{1}{\beta}\, \log \phi_\beta$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when $\log g(x)=-\log f'(x)$. In that case we relate the involution kernel to the so called scaling function.