Creating pseudo-Anosov Maps from Permutations and Matrices
Abstract
We provide an integral combinatorial characterization of pseudo-Anosov maps on closed oriented surfaces of genus g > 1. We show that an orientation-preserving pseudo-Anosov homeomorphism with orientable foliations and fixing all critical trajectories can be encoded as a permutation of 2g+v-1 positive integers, where v is the number of singular points of the foliations (disregarding multiplicity). We call such a permutations an ordered block permutation (OBP), and it satisfies an admissiblity condition. Conversely, we show that a surface along with measured foliations (up to scaling) and the pseudo-Anosov map can be uniquely constructed out of the data of an admissible permutation of 2g+v-1 positive integers. In particular, for closed surfaces, we construct every orientable foliation invariant under a pseudo-Anosov homeomorphism.
Cite
@article{arxiv.1902.07440,
title = {Creating pseudo-Anosov Maps from Permutations and Matrices},
author = {John H. Hubbard and Ahmad Rafiqi and Tom Schang},
journal= {arXiv preprint arXiv:1902.07440},
year = {2024}
}