Path-connectivity of Thick Laminations, and Markov Processes with Thick Limit Sets

Abstract

A lamination $\hat{I}»$ is $\hat{I}\mu$-thick (with respect to a basepoint $X$), if the Teichmüller ray from $X$ in the direction of $\hat{I}»$ stays in the $\hat{I}\mu$-thick part. We show that, for surfaces of high enough genus, any two $\hat{I}\mu$-thick laminations can be joined by a path of $\hat{I}´$-thick laminations. As a consequence, we show that the Morse boundary of the mapping class group is path-connected. Furthermore, we construct a subshift of finite type on the mapping class group, whose limit set consists only of thick laminations and is path-connected.

30 pages