Constructing pseudo-Anosov maps with given dilatations

Abstract

In this paper, we give sufficient conditions for a Perron number, given as the leading eigenvalue of an aperiodic matrix, to be a pseudo-Anosov dilatation of a compact surface. We give an explicit construction of the surface and the map when the sufficient condition is met.

An algorithm to compute the Teichmüller polynomial from matrices

“…In this case, the curves connecting Q i and Q i+1 in P 0 lift to loops which form a basis of H 1 (S g ). It is also observed in [2] that with respect to this basis, M represents the action of the lift of the pseudo-Anosov map constructed above on H 1 (S g ).…”

“…In this section, we quickly recall the construction of surfaces given in [2]. We start with an aperiodic non-singular n × n matrix M with only entires 0 and 1 satisfying so-called odd-block condition.…”

“…Note that the odd-block in some column could be empty. The concept of oddblock matrices are originally introduced in [16], but the name was coined in [2] where the properties of odd-block matrices were further investigated.…”

“…In the rest of the section, we will follow the recipe given in [2] to construct a surface of finite type and an orientation-preserving pseudo-Anosov homeomorphism from the above matrix M . In fact, one could construct an orientation-reserving homeomorphism but since we are going to use only orientation-preserving homeomorphisms for the rest of the paper, we only recall the process of constructing orientation-preserving pseudo-Anosov homeomorphisms.…”

“…In this paper, we present a simple algorithm to compute Teichmüller polynomial for so-called odd-block surfaces constructed from some {0, 1}matrices in a precedent work of the authors [2]. The definition and construction of such surfaces will be recalled in Section 2.…”