Entropy, Weil-Petersson translation distance and Gromov norm for surface automorphisms

TL;DR

This paper establishes linear bounds relating the entropy and Weil-Petersson translation distance of surface automorphisms to the Gromov norm of their mapping tori, linking geometric and dynamical properties.

Contribution

It proves new linear bounds connecting entropy, Weil-Petersson translation distance, and Gromov norm for surface automorphisms, extending Brock's theorem.

Findings

Entropy has linear bounds in terms of Gromov norm.

Weil-Petersson translation distance is similarly bounded from both sides.

Results are derived from Brock's comparison theorem and additional geometric theorems.

Abstract

Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. We also prove that the Weil-Petersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock together with some other strong theorems and small observations.