Regular blocks and Conley index of isolated invariant continua in surfaces

TL;DR

This paper investigates the topological and dynamical properties of isolated invariant continua on surfaces, establishing their shape, regularity, and Conley index, and classifying certain invariant sets based on their dynamics.

Contribution

It introduces the concept of regular isolating blocks, computes Conley indices using cohomology ring structures, and classifies invariant continua without fixed points.

Findings

Isolated invariant continua in surfaces have the shape of finite polyhedra.

Existence of regular isolating blocks allows for Conley index computation.

The cohomology ring structure determines the Conley index.

Abstract

In this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces. We show that near an isolated invariant continuum the flow is topologically equivalent to a C1 flow. We deduce that isolated invariant continua in surfaces have the shape of finite polyhedra. We also show the existence of regular isolating blocks of isolated invariant continua and we use them to compute their Conley index provided that we have some knowledge about the truncated unstable manifold. We also see that the ring structure of the cohomology index of an isolated invariant continuum in a surface determines its Conley index. In addition, we study the dynamics of non-saddle sets, preservation of topological and dynamical properties by continuation and we give a topological classification of isolated invariant continua which do not contain fixed points and, as a consequence, we also classify isolated minimal sets.