Slow north-south dynamics on \(\mathcal{PML}\) (Q2406849)

\(S_{g,p}\) denotes the surface of genus \(g\) and \(p\) punctures. The authors state the following theorem that clarifies a claim by W. Thurston. Theorem. Suppose that \(3g-3+p \geq 4\) and fix a finite generating set for \({\mathrm{Mod}}^+(S_{p,q})\). There exists an infinite family of pseudo-Anosov mapping classes where the rate of convergence goes to one, and decays exponentially with respect to the word length. In the second section of the paper, the authors show how to construct a pseudo-Anosov family on \(S_{3,0}\) whose spectral ratio goes to one exponentially with the word length. This construction is extended to other surfaces. The possible spectral ratios of pseudo-Anosov mapping classes are summarized in Table 1. For example, the aforementioned table shows that there are no pseudo-Anosov mapping classes for \(S_{0,0}\), \(S_{0,1}\),\(S_{0,2}\) and \(S_{0,3}\). For \(S_{0,4}\) the spectral ratios of pseudo-Anosov mapping classes are bounded away from one. In the last section the authors use the Python package Flipper for a rigorous verification of the constructed examples.