Entropy vs volume for pseudo-Anosov maps

2008/12/15 by Eiko Kin, Sadayoshi Kojima, Kin, Eiko +3
Mathematics · #37E30 #57M27 #57M55 #Advanced Combinatorial Mathematics #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DS #math.GT #msc:37E30 #msc:57M27 #msc:57M55

paper · pdf · doi:10.48550/arxiv.0812.2941

16 pages, 14 figures

arxiv created 2008/12/15 · openalex publication_date 2008/12/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.

Citations

• Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle

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• Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori
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