A Transcendental Invariant of Pseudo-Anosov Maps

For each pseudo-Anosov map $\varphi$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,\varphi )$. $A(S,\varphi )$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov maps. We will develop a few nice properties of $A(S,\varphi )$ and give a few examples to show that $A(S,\varphi )$ 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.