2014/11/24 by Sadayoshi Kojima, Greg McShane
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Automorphism #Combinatorics #Entropy (arrow of time) #Geometric and Algebraic Topology #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Pure mathematics #Statistical physics #Statistics #Surface (topology) #Topological and Geometric Data Analysis #Topological entropy #Torus #Upper and lower bounds #math.GT #msc:37E30 #msc:57M27 #msc:57M55
paper · pdf · doi:10.2140/gt.2018.22.2403
published as Geom. Topol. 22 (2018) 2403-2426 · 17 pages, 2 figures Updated version with expanded section on the duality pairing betweeen Beltrami differentials and quadratic forms and corrections to the exposition of Nehari's Theorem
openalex publication_date 2014/11/24 · arxiv created 2015/02/27 · arxiv updated 2018/04/18 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As its corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.