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

A lamination $\lambda$ is $\epsilon$-thick (with respect to a basepoint $X$), if the Teichmüller ray from $X$ in the direction of $\lambda$ stays in the $\epsilon$-thick part. We show that, for surfaces of high enough genus, any two $\epsilon$-thick laminations can be joined by a path of $\delta$-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.