Uniform rigidity sequences for weak mixing diffeomorphisms on $\mathbb{T}^2$

TL;DR

This paper demonstrates that for certain growth rate sequences, one can construct weak mixing diffeomorphisms on the 2-torus that are uniformly rigid with respect to those sequences, using a quantitative Anosov-Katok method.

Contribution

It introduces a new approach to constructing weak mixing diffeomorphisms with prescribed uniform rigidity sequences using explicit conjugation maps in smooth and real-analytic topologies.

Findings

Constructed weak mixing diffeomorphisms with specified rigidity sequences.

Extended the Anosov-Katok method to a quantitative framework.

Achieved results in both $C^{ abla}$-topology and real-analytic topology.

Abstract

In this paper we will show that if a sequence of natural numbers satisfies a certain growth rate, then there is a weak mixing diffeomorphism on $\mathbb{T}^2$ that is uniformly rigid with respect to that sequence. The proof is based on a quantitative version of the Anosov-Katok-method with explicitly defined conjugation maps and the constructions are done in the $C^{\infty}$-topology as well as in the real-analytic topology.