Pseudo-Anosov maps with small stretch factors on punctured surfaces

Abstract

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behavior of this minimum number for a certain large subset of the $\left(g,n\right)$ plane, up to a multiplicative constant. In particular, we show that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.