A Remark to the Theorem of Le Calvez and Yoccoz

TL;DR

This paper explores the limitations of the Le Calvez and Yoccoz theorem on minimal homeomorphisms, showing it does not extend to other surfaces and analyzing the failure of the fast-conjugation method.

Contribution

It demonstrates the theorem's inapplicability to surfaces beyond the punctured sphere and discusses the shortcomings of the fast-conjugation method in constructing minimal homeomorphisms.

Findings

The theorem does not hold for other surfaces.

The fast-conjugation method often fails to construct minimal homeomorphisms.

The article revisits an old unpublished work with corrected insights.

Abstract

The theorem of Le Calvez and Yoccoz states that there are no minimal homeomorphisms on the finite punctered 2-dimensional sphere S 2 . We show that this does not hold for other surfaces. Moreover, we discuss why the fast-conjugation-method fails in the most cases to construct such homeomorphisms. This article based on an old unpublished article (Quasi-Minimal, Pseudo-Minimal Systems and Dense Orbits) with incorrect results.