Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs

TL;DR

This paper proves that for certain infinite Riemann surfaces with unbounded cuffs, appropriate twisting can induce ergodic geodesic flows, extending previous results and confirming a conjecture by Kahn and Markovi7c.

Contribution

It demonstrates that for any sequence of cuff lengths, suitable twists can make the surface parabolic, confirming a conjecture and extending results to surfaces with countably many ends.

Findings

Twists can induce parabolicity regardless of cuff lengths.

The result applies to surfaces with countably many ends.

The conjecture by Kahn and Markovi7c is essentially true.

Abstract

A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $\alpha_n$ of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths $\ell (\alpha_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (\alpha_n)$ (think of $\ell (\alpha_n)=n!^{n!}$) would allow the geodesic flow to escape to infinity, thus making $X$ not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Markovi\'c conjectured that for every choice of lengths $\ell (\alpha_n)$, there is a choice of twists that would make $X$ parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers $\{ a_n\}$, there is a choice of lengths $\ell (\alpha_n)\geq a_n$ such that the (relative) twists by $1/2$ make $X$ parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends.