A transcendental invariant of pseudo-Anosov maps

For each pseudo-Anosov map phi : S -> S, we associate it with a Q-vector space lying in R, and denote it by A(S,phi). The invariant A(S,phi) is defined by an interaction between the Thurston norm and the dilatation of pseudo-Anosov maps. We develop a few nice properties of A(S,phi) and give a few examples to show that A(S,phi) is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen, and show that the minimal point of the restriction of the dilatation function on the fibered face need not be a rational point.