Proofs of ergodicity of piecewise M\"obius interval maps using planar extensions

TL;DR

This paper establishes new criteria for deducing ergodic and expansive properties of piecewise Möbius interval maps from their planar extensions, relaxing traditional conditions and applying quilting techniques.

Contribution

It introduces a new property 'bounded non-full range' for planar extensions and demonstrates how quilting can infer dynamical properties of interval maps.

Findings

Eventual expansivity and ergodic measures follow from finiteness conditions.

The quilting operation can prove key dynamical properties.

Results recover known properties for Nakada α-continued fractions and extend to non-commensurable Fuchsian groups.

Abstract

We give two results for deducing dynamical properties of piecewise M\"obius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property ``bounded non-full range" used to relax traditional Markov conditions. Second, the ``quilting" operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise M\"obius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $\alpha$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.