Smooth Conjugacy classes of circle diffeomorphisms with irrational rotation number

TL;DR

This paper proves that within the set of circle diffeomorphisms with a fixed irrational rotation number, each conjugacy class is dense in the $C^1$ topology, highlighting the richness of conjugacy classes.

Contribution

It establishes the $C^1$-density of all $C^r$-conjugacy classes for circle diffeomorphisms with a specified irrational rotation number.

Findings

Every $C^r$-conjugacy class is dense in the $C^1$ topology.

The result applies to all diffeomorphisms with a fixed irrational rotation number.

The proof advances understanding of the structure of circle diffeomorphisms.

Abstract

In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.