Doubly Intermittent Maps with Critical Points, Unbounded Derivatives and Regularly Varying Tail

TL;DR

This paper studies a class of complex dynamical systems with critical points and unbounded derivatives, establishing conditions for the existence and finiteness of a unique invariant measure, extending previous work to include regularly varying tails.

Contribution

It extends prior research by analyzing doubly intermittent maps with critical points and unbounded derivatives, focusing on the role of regularly varying tails in measure finiteness.

Findings

Existence of a unique mixing absolutely continuous invariant measure.

Conditions for the measure to be finite or infinite based on tail behavior.

Extension of previous results to maps with regularly varying tails.

Abstract

We consider a class of doubly intermittent maps with critical points, unbounded derivative and regularly varying tails. Under some mild assumptions we prove the existence of a unique mixing absolutely continuous invariant measure and give conditions under which the measure is finite. This extends former work by Coates, Luzzatto and Mubarak to maps with regularly varying tails. Particularly, we look at the boundary case where the behaviour of the slowly varying function decides if the invariant measure is finite or infinite.