The Local Rotation Set is an Interval

TL;DR

This paper proves that the local rotation set for homeomorphisms of the plane isotopic to the identity, fixing a point, is always an interval, confirming a conjecture and extending results to the open annulus.

Contribution

It establishes that the local rotation set is always an interval for homeomorphisms of the plane and the open annulus, answering a previously open question.

Findings

The local rotation set is always an interval for plane homeomorphisms.

The result extends to the case of the open annulus.

Confirms a conjecture posed by Frédéric Le Roux.

Abstract

Let $Homeo\_0 (R 2 ; 0)$ be the set of all homeomorphisms of the plane isotopic to the identity and which fix 0. Recently in the article entitled "L'ensemble de rotation local autour d'un point fixe" Fr{\'e}d{\'e}ric Le Roux gave the definition of the local rotation set of an isotopy in $Homeo\_0 (R 2 ; 0)$ from the identity to a homeomorphism f and he asked if this set is always an interval. In this article we give a positive answers to this question and to the analogous question in the case of the open annulus.