On the asymptotics of the $\alpha$-Farey transfer operator

TL;DR

This paper investigates the long-term behavior of the transfer operator for certain non-uniformly hyperbolic maps, identifying conditions under which asymptotic properties hold or fail, with implications for understanding dynamical systems.

Contribution

It introduces a family of observables with specific properties for which the transfer operator's iterates do not follow expected asymptotics, and provides conditions ensuring the asymptotic behavior is consistent across the space.

Findings

Identified observables where transfer operator iterates deviate from expected asymptotics.

Established conditions under which the asymptotic behavior of transfer operator iterates is consistent.

Extended theorems linking asymptotic behavior on initial partition elements to broader sets.

Abstract

We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the iterates of the transfer operator, when applied to one of these observables, is not asymptotic to a constant times the wandering rate on the first element of the partition $\alpha$. Subsequently, sufficient conditions on observables are given under which this expected asymptotic holds. In particular, we obtain an extension theorem which establishes that, if the asymptotic behaviour of iterates of the transfer operator is known on the first element of the partition $\alpha$, then the same asymptotic holds on any compact set bounded away from the indifferent fixed point.