A Transcendental Invariant of Pseudo-Anosov Maps
arXiv:1209.2613 · doi:10.1112/jtopol/jtv010
Abstract
For each pseudo-Anosov map $Ï$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,Ï)$. $A(S,Ï)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov maps. We will develop a few nice properties of $A(S,Ï)$ and give a few examples to show that $A(S,Ï)$ is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen: the minimal point of the restriction of the dilatation function on fibered face need not be a rational point.
32 pages, 10 figures, abstract has been modified by following suggestion from Curtis McMullen