Existence of torsion-low maximal identity isotopies for area preserving surface homeomorphisms

TL;DR

This paper proves the existence of special maximal isotopies called torsion-low isotopies for area-preserving surface homeomorphisms, enabling detailed analysis of their local dynamics and transverse foliations.

Contribution

It introduces torsion-low isotopies and establishes their existence as maximal identity isotopies for area-preserving surface homeomorphisms.

Findings

Existence of torsion-low maximal identity isotopies.

Characterization of torsion-low isotopies via rotation numbers.

Description of local dynamics near isolated singularities.

Abstract

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $\rho(I,z)$, which is well defined when one blows-up $f$ at $z$, is contained in $(-1,1)$. We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity.