An index theory for transverse trajectories

TL;DR

This paper introduces a new index theory for Brouwer homeomorphisms, extending classical concepts to broader settings and linking indices with transverse foliations, thus advancing topological dynamics understanding.

Contribution

It provides an alternative definition of the Le Roux index that connects Brouwer homeomorphism indices with transverse foliation structures, answering an open question.

Findings

New index definition for Brouwer homeomorphisms

Established link between indices and transverse foliations

Generalized Poincaré-Hopf index to non-singular planar flows

Abstract

In this work, we present an alternative definition of the Le Roux index, which generalizes the Poincar\'e-Hopf index for non-singular planar flows to the broader setting of Brouwer homeomorphisms. This new approach answers a question raised by Le Roux by establishing a connection between the index of a Brouwer homeomorphism and the structure of its transverse foliations, in the sense of Le Calvez.