On the Thermodynamic Formalism for the Farey Map

TL;DR

This paper analyzes the Farey map's intermittency using thermodynamic formalism, employing induced systems to study zeta functions and pressure, revealing phase transitions and asymptotic behaviors.

Contribution

It introduces an induced subsystem approach to overcome convergence issues caused by an indifferent fixed point, linking zeta functions and transfer operators for the Farey map.

Findings

Analytic properties of the induced zeta function are established.

Asymptotic behavior of the pressure function as temperature tends to zero is characterized.

Existence of a phase transition at inverse temperature β=1 is demonstrated.

Abstract

The chaotic phenomenon of intermittency is modeled by a simple map of the unit interval, the Farey map. The long term dynamical behaviour of a point under iteration of the map is translated into a spin system via symbolic dynamics. Methods from dynamical systems theory and statistical mechanics may then be used to analyse the map, respectively the zeta function and the transfer operator. Intermittency is seen to be problematic to analyze due to the presence of an `indifferent fixed point'. Points under iteration of the map move away from this point extremely slowly creating pathological convergence times for calculations. This difficulty is removed by going to an appropriate induced subsystem, which also leads to an induced zeta function and an induced transfer operator. Results obtained there can be transferred back to the original system. The main work is then divided into two sections. The first demonstrates a connection between the induced versions of the zeta function and the transfer operator providing useful results regarding the analyticity of the zeta function. The second section contains a detailed analysis of the pressure function for the induced system and hence the original by considering bounds on the radius of convergence of the induced zeta function. In particular, the asymptotic behaviour of the pressure function in the limit $\beta$, the inverse of `temperature', tends to negative infinity is determined and the existence and nature of a phase transition at $\beta=1$ is also discussed.