Topological entropy of pseudo-Anosov mapping classes on punctured surfaces

Saadet Öykü Yurttaş
Dicle Üniversitesi, Türkiye

We present a fast method for computing the topological entropy of pseudo-Anosov mapping classes on a finitely punctured surface S. The method is based on results in Thurston’s work on surface homeomorphisms, and makes use of matrices that can be viewed as generalizations of Dynnikov matrices for pseudo-Anosov braids on the finitely punctured disk. More precisely, the method associates to a given pseudo-Anosov mapping class f in S a matrix which describes the action of f near its invariant unstable measured foliation on the space of projective measured foliations of S. The leading eigenvalue of such matrices gives the topological entropy of f. If time permits, we compare the spectra of such matrices with the spectra of train track transition matrices of a given pseudo-Anosov mapping class, and show that these matrices are isospectral up to roots of unity and zeros under some suitable conditions.