On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps

TL;DR

This paper investigates how unbounded observables with singularities converge to equilibrium under a family of intermittent interval maps, revealing that the limit behavior depends on the structure of the omega-limit set.

Contribution

It introduces a detailed analysis of the convergence behavior of unbounded observables with singularities under a continuum of maps interpolating between Tent and Farey maps, highlighting the dependence on omega-limit sets.

Findings

Convergence behavior varies with the structure of the omega-limit set.

Unbounded observables with singularities do not fit into previously studied classes for convergence.

The asymptotic behavior of Perron-Frobenius iterates is characterized for these observables.

Abstract

We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $\beta \in [0, 1]$ and $\alpha \in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $\beta$ of order $\alpha$ is dependent on the structure of the $\omega$-limit set of $\beta$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function classes on which convergence to equilibrium has been previously shown.