  Within $N$-Calabi-Yau categories associated with quivers whose base graphs
form trees, we delve into the study of the asymptotic behaviors of
autoequivalences of a specific type. These autoequivalences, which we call
"Penner type," exhibit straightforward asymptotic characteristics, making them
noteworthy exemplars of "pseudo-Anosov" autoequivalences in the sense of
\cite{Fan-Filip-Haiden-Katzarkov-Liu21}, and also in a stronger sense that we
define in the present paper.
  In addition, we provide a practical methodology for calculating the
stretching factors of Penner type autoequivalences. We expect that this
computational approach can have applications. As an example, we establish a
positive lower bound on the translation length of the induced action these
autoequivalences have on the space of stability conditions. Our anticipation is
that this lower bound is, in fact, exact. Notably, we have observed instances
of Penner type $\Phi$ where the induced actions align precisely with this lower
bound. In other words, these examples induce hyperbolic actions on the space of
stability conditions.
