Small dilatation pseudo-Anosov mapping classes and short circuits on train track automata

Eriko Hironaka

This note is a survey of recent results surrounding the minimum dilatation problem for pseudo-Anosov mapping classes. In particular, we give evidence for the conjecture that the minimum accumulation point of the genus normalized dilatations of pseudo-Anosov mapping classes on closed surfaces equals the square of the golden ratio. We also find explicit fat train track maps determining a sequence of pseudo-Anosov mapping classes whose normalized dilatations converge to this limit.