Let $S_g$ be the closed surface of genus $g$, $\mathcal{ L}$ be the infinite Jacob's ladder surface, and $\mathrm{ Map}(S)$ denote the mapping class group of a surface $S$. Let $q_g:\mathcal{ L}\to S_g$ be the regular infinite-sheeted cover with deck transformation group $\mathbb{ Z}$. In this paper, we show the existence of ``pseudo-Anosov-like'' maps on $\mathcal{ L}$ that arise as the lifts of Penner-type pseudo-Anosov maps on $S_g$ under the cover $q_g$. Furthermore, we establish that these lifts are topologically transitive, mixing, and support null recurrent dynamics. Moreover, we present concrete examples of infinite families of such maps on $\mathcal{ L}$.
Categories: math.GT math.DS math.PR