On a family of pseudo-Anosov-like maps on the infinite ladder surface

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 $\mathcal{L}$.