# Real-analytic realization of Uniform Circular Systems and some applications

## TL;DR

This paper demonstrates that uniform circular systems can be realized as real-analytic diffeomorphisms on the torus, extending anti-classification results to real-analytic ergodic systems and constructing many non-Kakutani equivalent examples.

## Contribution

It provides the first real-analytic realizations of uniform circular systems on the torus, linking ergodic theory with real-analytic dynamics.

## Key findings

- Real-analytic realizations of uniform circular systems on the torus.
- Anti-classification results for real-analytic ergodic diffeomorphisms.
- Existence of uncountably many non-Kakutani equivalent real-analytic diffeomorphisms.

## Abstract

Recently Matthew Foreman and Benjamin Weiss showed in a series of papers that smooth ergodic diffeomorphisms of a compact manifold are unclassifiable up to measure-isomorphism. In this paper we show that the uniform circular systems used in the work of Foreman-Weiss admit real-analytic realizations on the torus. As a consequence we obtain the same anti-classification result for real-analytic ergodic diffeomorphisms on the torus. In another application we show the existence of an uncountable family of pairwise non-Kakutani equivalent real-analytic diffeomorphisms on the torus.