Transience and thermodynamic formalism for infinitely branched interval maps

TL;DR

This paper explores the dynamics and thermodynamic formalism of a family of countably piecewise linear interval maps, analyzing phase transitions between conservative and dissipative behaviors and computing associated pressures.

Contribution

It introduces a detailed analysis of phase transitions and pressure functions in countably piecewise linear maps that lack the large image property, extending thermodynamic formalism.

Findings

Identification of second order phase transitions in the pressure function

Explicit computation of various pressure definitions using recurrence relations

Characterization of conservative and dissipative behaviors in the family of maps

Abstract

We study a one-parameter family of countably piecewise linear interval maps, which, although Markov, fail the `large image property'. This leads to conservative as well as dissipative behaviour for different maps in the family with respect to Lebesgue. We investigate the transition between these two types, and study the associated thermodynamic formalism, describing in detail the second order phase transitions (i.e. the pressure function is $C^1$ but not $C^2$ at the phase transition) that occur in transition to dissipativity. We also study the various natural definitions of pressure which arise here, computing these using elementary recurrence relations.