[Paper Review] Normalized Entropy versus Volume

Sadayoshi Kojima, Greg McShane|arXiv (Cornell University)|Nov 24, 2014

Mathematical Dynamics and Fractals5 citations

TL;DR

This paper establishes an explicit linear inequality between the normalized entropy of pseudo-Anosov mapping classes and the hyperbolic volume of their mapping tori, proving that normalized entropy is bounded below by a universal constant times volume. Using results from quasi-Fuchsian geometry and geometric finiteness, the authors derive sharp lower bounds for entropy and prove a finiteness result for manifolds with small normalized entropy, improving on prior work by Penner, Farb-Leininger-Margalit, and Agol.

ABSTRACT

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.

Motivation & Objective

• To establish an explicit, sharp lower bound for the normalized entropy of pseudo-Anosov mapping classes in terms of the hyperbolic volume of their mapping tori.
• To improve upon Penner's lower bound for entropy on surfaces with punctures by incorporating the minimal volume of cusped hyperbolic 3-manifolds.
• To provide a new proof of the finiteness theorem for cusped hyperbolic 3-manifolds generating pseudo-Anosovs with small normalized entropy, as previously shown by Farb-Leininger-Margalit and Agol.
• To extend the entropy-volume inequality to the Weil-Petersson translation distance, yielding a stronger relationship between geometric invariants.

Proposed method

• Leverage Schlenker's recent results on renormalized volumes of quasi-Fuchsian manifolds to derive an intermediate inequality between convex core volume and Teichmüller translation distance.
• Apply geometric finiteness and geometric inelasticity (from Brock-Bromberg and Minsky) to analyze the structure of the convex core in quasi-Fuchsian manifolds associated with iterated mapping classes.
• Use bilipschitz maps from the deep part of the convex core to the infinite cyclic cover of the mapping torus to estimate volume growth via the number of translates of a fundamental domain.
• Estimate the number of closed geodesics in the deep part using Minsky's model and entropy-based depth bounds, leading to a linear lower bound in terms of the entropy.
• Cover the convex core with a thick part (Vn) and a thin part (NT(∂Cn)) to derive upper and lower volume bounds, using uniform diameter estimates and geometric series decay.
• Replace Teichmüller translation distance with Weil-Petersson translation distance in the argument to obtain a stronger inequality, yielding a better constant in the volume bound.

Experimental results

Research questions

• RQ1 Can an explicit, sharp lower bound be established for the normalized entropy of pseudo-Anosovs in terms of the hyperbolic volume of their mapping tori?
• RQ2 Does the finiteness result for cusped hyperbolic 3-manifolds generating small normalized entropy pseudo-Anosovs hold with an explicit, computable constant?
• RQ3 Can the relationship between entropy and volume be strengthened by replacing Teichmüller translation distance with Weil-Petersson translation distance?
• RQ4 What is the minimal volume of a hyperbolic 3-manifold that fibers over the circle with a given surface topology, and how does it relate to entropy?

Key findings

• The paper proves the inequality $\text{ent } \phi \geq \frac{1}{3\pi|\chi(\Sigma)|} \text{vol } N_\phi$, establishing a universal lower bound for normalized entropy independent of surface topology.
• For surfaces with at least one puncture, the entropy satisfies $\text{ent } \phi \geq \frac{2v_3}{3\pi(2g - 2 + m)}$, where $v_3 \approx 1.01494$ is the volume of the regular ideal simplex.
• The figure-eight knot complement achieves the conjectured minimal entropy-to-volume ratio of approximately 0.47412, suggesting the bound is close to sharp.
• The authors prove a slightly weaker version of the Farb-Leininger-Margalit finiteness theorem, showing that only finitely many cusped hyperbolic 3-manifolds can generate pseudo-Anosovs with bounded normalized entropy.
• By replacing Teichmüller distance with Weil-Petersson translation distance, the paper obtains a stronger inequality: $||\phi||_{WP} \geq \frac{2}{3} \sqrt{2\pi|\chi(\Sigma)|} \cdot \text{vol } N_\phi$, which improves the volume bound in Brock's theorem.
• The Weil-Petersson distance relationship is shown to be stronger than the Teichmüller distance one, as there exist sequences of pseudo-Anosovs with bounded Weil-Petersson distance but divergent entropy, indicating a more robust geometric link.