A H\"older-type inequality for the $C^0$ distance and Anosov-Katok pseudo-rotations

TL;DR

This paper establishes a H"older-type inequality linking various norms of Hamiltonian diffeomorphisms, demonstrating that rapid convergence in Hofer/spectral metrics implies $C^0$ convergence, and applies this to pseudo-rotations from the Anosov-Katok method.

Contribution

It introduces a new H"older-type inequality connecting $C^0$, derivative, and Hofer/spectral norms for Hamiltonian diffeomorphisms, and applies it to prove $C^0$ rigidity for certain pseudo-rotations.

Findings

Fast convergence in Hofer/spectral metric implies $C^0$ convergence.

Proves a $C^0$ rigidity result for Anosov-Katok pseudo-rotations.

Establishes a H"older-type inequality relating key norms.

Abstract

We prove a H\"older-type inequality for Hamiltonian diffeomorphisms relating the $C^0$ norm, the $C^0$ norm of the derivative, and the Hofer/spectral norm. We obtain as a consequence that sufficiently fast convergence in Hofer/spectral metric forces $C^0$ convergence. The second theme of our paper is the study of pseudo-rotations that arise from the Anosov-Katok method. As an application of our H\"older-type inequality, we prove a $C^0$ rigidity result for such pseudo-rotations.