Generalized Farey trees, transfer Operators and phase transitions

TL;DR

This paper studies a family of Markov maps that interpolate between the tent and Farey maps, analyzing their dynamics via transfer operators, revealing phase transitions and unique properties relevant to number theory.

Contribution

It introduces a generalized family of maps combining properties of tent and Farey maps and analyzes their spectral and thermodynamic characteristics.

Findings

Identification of first and second order phase transitions.

Discovery of positivity in the interaction function.

Spectral analysis of transfer operators for the family.

Abstract

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter map is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral analysis of generalized transfer operators. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.