Real-analytic AbC constructions on the torus

TL;DR

This paper extends the AbC method to real-analytic diffeomorphisms on the torus, enabling new constructions of minimal, non-uniquely ergodic, and nonstandard toral translations with real-analytic regularity.

Contribution

It introduces a general framework for real-analytic AbC constructions on the torus and demonstrates their application to novel dynamical systems.

Findings

Constructed minimal but not uniquely ergodic diffeomorphisms

Real-analytic realizations of nonstandard toral translations

Extended AbC method from smooth to real-analytic category

Abstract

In this article we demonstrate a way to extend the AbC (approximation by conjugation) method invented by Anosov and Katok from the smooth category to the category of real-analytic diffeomorphisms on the torus. We present a general framework for such constructions and prove several results. In particular, we construct minimal but not uniquely ergodic diffeomorphisms and nonstandard real-analytic realizations of toral translations.