The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of the minimal pseudo-Anosovs dilatations

TL;DR

This paper investigates the asymptotic behavior of minimal pseudo-Anosov dilatations on surfaces, establishing an upper bound for their growth rate under specific topological conditions.

Contribution

It proves a new upper bound on the growth rate of minimal dilatations for certain families of surfaces, extending understanding of pseudo-Anosov dynamics.

Findings

Established that $rac{n \, log \, ext{dilatation}}{log n} o ext{bounded by 2}$ under specified conditions.

Connected the boundary of fibered faces of the magic 3-manifold to dilatation asymptotics.

Provided conditions on genus and punctures for the asymptotic upper bound.

Abstract

Let $\delta_{g,n}$ be the minimal dilatation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. Tsai proved that for any fixed $g \ge 2$, the logarithm of the minimal dilatation $\log \delta_{g,n}$ is on the order of $\frac{\log n}{n}$. The main result of this paper is that if $2g+1$ is relatively prime to $s$ or $s+1$ for each $0 \le s \le g$, then $$\limsup_{n \to \infty} \frac{n \log \delta_{g,n}}{\log n} \le 2.$$