Regular blocks and Conley index of isolated invariant continua in surfaces. (November 2016)

• Record Type: Journal Article
• Authors: Barge, Héctor
• Is Part Of: Nonlinear analysis. Volume 146(2016)
• Page Start: 100
• Page End: 119
• Publication Date: 2016-11
• Subjects: 34C45 -- 37G35 -- 58J20; Conley index -- Regular isolating block -- Unstable manifold -- Fixed point -- Minimal set -- Non-saddle set; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.08.023
• Languages: English
• ISSNs: 0362-546X

Abstract

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 C 1 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 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.