2019/02/20 by John H. Hubbard, Hubbard, John H., Ahmad Rafiqi +3
Computer Science · Mathematics · #Algebraic Topology (math.AT) #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1902.07440
openalex publication_date 2019/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.