Non-standard real-analytic realizations of some rotations of the circle

TL;DR

This paper develops a real-analytic approximation method to construct specific zero entropy, uniquely ergodic diffeomorphisms of the torus that are metrically isomorphic to certain irrational circle rotations, expanding the understanding of such dynamical systems.

Contribution

It introduces a real-analytic version of the smooth approximation by conjugation method to realize specific circle rotations on the torus.

Findings

Constructed real-analytic diffeomorphisms with zero entropy

Demonstrated isomorphism to Liouvillian irrational rotations

Extended approximation techniques to real-analytic setting

Abstract

We extend some aspects of the smooth approximation by conjugation method to the real-analytic set-up and create examples of zero entropy, uniquely ergodic real-analytic diffeomorphisms of the two dimensional torus metrically isomorphic to some (Liouvillian) irrational rotations of the circle.