Thermodynamic formalism for transient dynamics on the real line

TL;DR

This paper introduces a new thermodynamic formalism to analyze the transient dynamics of skew-periodic extensions of expanding maps on the real line, revealing a dimension gap linked to non-zero drift.

Contribution

The work develops a novel thermodynamic approach for studying transient behaviors and dimension spectra in skew-periodic real line maps, connecting to Kleinian group phenomena.

Findings

Dimension gap occurs with non-zero drift

Precise quantification of the dimension gap

Model for Kleinian group limit set phenomena

Abstract

We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic $\mathbb{Z}$-extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the whole dimension spectrum with respect to $\alpha$-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.