Regular blocks and Conley index of isolated invariant continua in surfaces

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.

Omega limits, prolongational limits and almost periodic points of a continuous flow via exterior spaces

“…In particular, our techniques based on exterior flows can be used to generalize some results obtained in [1][2][3].…”

“…We also suggest some future research lines: (1) the higher homotopy invariant of exterior spaces (Steenrod,Čech and Brown-Grossmann groups) could be very useful tools in the analysis of local and global stability properties of continuous flows and in the study of chaotic dynamical systems; (2) we think that the externologies ε (ǫ,τint) (X) will play an important role in the study of repellers and atractors of a dynamical system, where τ int is the intrinsic topology of a flow X.…”

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

“…This is motivated by some results from [23,24] about certain unstable attractors. In addition, motivated by the results in [6] and [3] about the continuation properties of isolated non-saddle sets we find necessary and sufficient conditions for the property of being non-saddle to be robust for families of smooth flows defined on smooth manifolds without further assumptions about the dimension or cohomology of the phase space.…”

“…More specifically, small perturbations of a flow preserve attractors and some of their basic topological properties [25] while small perturbations may transform isolated non-saddle sets into saddle ones with distinct topological structures [13]. However, as established in [3,6] there are some situations in which the robustness of some topological properties is equivalent to the robustness of non-saddleness.…”

“…However, it turns out that there exist some relations between the preservation of certain topological properties by continuation and the preservation of the dynamical property of non-saddleness. For instance, if the phase space is a surface [3,Theorem 26] or a closed orientable smooth manifold with trivial integral first cohomology group [6,Theorem 39], the property of being non-saddle is preserved by continuation if and only if the shape is preserved.…”