Chain Recurrent Research Articles

Recurrent sets for endomorphisms of topological groups

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, we show that some dynamical properties are induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property.

Chain recurrence and positive shadowing in linear dynamics

We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even on normed vector spaces. We show that for linear operators there is only one chain recurrent set, and this set is a closed invariant subspace. We prove that every chain transitive linear dynamical system with positive shadowing property is frequently hypercyclic and, as a corollary, we obtain that every positive shadowing hypercyclic linear dynamical system is frequently hypercyclic.

Chain recurrence and structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of multivalued semiflows

We study properties of \begin{document}$ \omega $\end{document} -limit sets of multivalued semiflows like chain recurrence or the existence of cyclic chains. First, we prove that under certain conditions the \begin{document}$ \omega $\end{document} -limit set of a trajectory is chain recurrent, applying this result to an evolution differential inclusion with upper semicontinous right-hand side. Second, we give conditions ensuring that the \begin{document}$ \omega $\end{document} -limit set of a trajectory contains a cyclic chain. Using this result we are able to check that the \begin{document}$ \omega $\end{document} -limit set of every trajectory of a reaction-diffusion equation without uniqueness of solutions is an equilibrium.

From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games.

Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set

ABSTRACTThe aim of this paper is to study in detail the relations between strong chain recurrence for flows and Lyapunov functions. For a continuous flow on a compact metric space, uniformly Lipschitz continuous on the compact subsets of the time, we first make explicit a Lipschitz continuous Lyapunov function strict – that is strictly decreasing – outside the strong chain recurrent set of the flow. This construction extends to flows some recent advances of Fathi and Pageault in the case of homeomorphisms; moreover, it improves Conley's result about the existence of a continuous Lyapunov function strictly decreasing outside the chain recurrent set of a continuous flow. We then present two consequences of this theorem. From one hand, we characterize the strong chain recurrent set in terms of Lipschitz continuous Lyapunov functions. From the other hand, in the case of a flow induced by a vector field, we establish a sufficient condition for the existence of a strict Lyapunov function and we also discuss various examples. Moreover, for general continuous flows, we show that the strong chain recurrent set has only one strong chain transitive component if and only if the only Lipschitz continuous Lyapunov functions are the constants. Finally, we provide a necessary and sufficient condition to guarantee that the strong chain recurrent set and the chain recurrent one coincide.

Chain Recurrence, Chain Transitivity, Lyapunov Functions and Rigidity of Lagrangian Submanifolds of Optical Hypersurfaces

The aim of this paper is twofold. On the one hand, we discuss the notions of strong chain recurrence and strong chain transitivity for flows on metric spaces, together with their characterizations in terms of rigidity properties of Lipschitz Lyapunov functions. This part extends to flows some recent results for homeomorphisms of Fathi and Pageault. On the other hand, we use these characterisations to revisit the proof of a theorem of Paternain, Polterovich and Siburg concerning the inner rigidity of a Lagrangian submanifold $$\Lambda $$ contained in an optical hypersurface of a cotangent bundle, under the assumption that the dynamics on $$\Lambda $$ is strongly chain recurrent. We also prove an outer rigidity result for such a Lagrangian submanifold $$\Lambda $$ , under the stronger assumption that the dynamics on $$\Lambda $$ is strongly chain transitive.

The generalized recurrent set and strong chain recurrence

Fathi and Pageault have recently shown a connection between Auslander’s generalized recurrent set$\text{GR}(f)$and Easton’s strong chain recurrent set. We study$\text{GR}(f)$by examining that connection in more detail, as well as connections with other notions of recurrence. We give equivalent definitions that do not refer to a metric. In particular, we show that$\text{GR}(f^{k})=\text{GR}(f)$for any$k>0$, and give a characterization of maps for which the generalized recurrent set is different from the ordinary chain recurrent set.

Spectral decomposition theorem in equicontinuous nonautonomous discrete dynamical systems

In this paper, we define the notion of weak chain recurrence and study properties of weak chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. Our main result is the Smale’s spectral decomposition theorem in an equicontinuous nonautonomous discrete dynamical system.

Some properties of chain recurrent sets in a nonautonomous discrete dynamical system

AbstractIn this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

Conley's theorem for dispersive systems

In this article, we study Conley's theorem about the chain recurrence in dynamical systems, that is, the chain recurrent set of continuous map $f$ is the complement of union of $B_{U}(A)-A$, where $A$ is an attractor and $B_{U}(A)$ is a basin of $A$. In this paper, we generalize this theorem to dispersive systems on noncompact spaces.

Chain Recurrences on Conservative Dynamics

Abstract. Let M be a manifold with a volume form ! and f : M ! M be a difieo-morphism of class C 1 that preserves ! . We prove that if M is almost bounded for thedifieomorphism f , then M is chain recurrent. Moreover, we get that Lagrange stablevolume-preserving manifolds are also chain recurrent. 1. IntroductionOur purpose of this paper is to study the chain recurrence set of volume-preserving difieomorphisms on non-compact manifolds. We follow Conley’s defl-nitions of attractors and chain recurrences [4], and Hurley’s generalized deflnitions[5],[6].From Poincar¶e recurrence theorem, it is well-known that for any volume-preserving difieomorphism on the compact manifolds M , every point of M is chainrecurrent. However, unfortunately, the parallel statement for the chain recurrencedoes not hold for the non-compact manifolds. Thus, in the non-compact case, wemay impose the canonical conditions as almost boundedness and Lagrange stability.Our main theorem is as follows. * Corresponding Author.

Chain recurrence in $\beta $-compactifications of topological groups

Let G be a topological group. In this paper limit behavior in the Stone–Čech compactification \beta G is studied. It depends on a family of translates of a reversible subsemigroup S . The notion of semitotal subsemigroup is introduced. It is shown that the semitotality property is equivalent to the existence of only two maximal chain transitive sets in \beta G whenever S is centric. This result links an algebraic property to a dynamical property. The concept of a chain recurrent function is also introduced and characterized via the compactification \beta G . Applications of chain recurrent function to linear differential systems and transformation groups are done.

Attractors and Chain Recurrence for Semigroup Actions

Let \({\mathcal{S}}\) be a semigroup acting on a topological space M. We define attractors for the action of \({\mathcal{S}}\) on M. This concept depends on a family \({\mathcal{F}}\) of subsets of \({\mathcal{S}}\) . For certain semigroups and families it recovers the concept of attractors for flows or semiflows. We define and study the complementary repeller of an attractor. We also characterize the set of chain recurrent points in terms of attractors.

Chain recurrence rates and topological entropy

We investigate the properties of chain recurrent, chain transitive, and chain mixing maps (generalizations of the well-known notions of non-wandering, topologically transitive, and topologically mixing maps). We describe the structure of chain transitive maps. These notions of recurrence are defined using ε-chains, and the minimal lengths of these ε-chains give a way to measure recurrence time (chain recurrence and chain mixing times). We give upper and lower bounds for these recurrence times and relate the chain mixing time to topological entropy.

On Strong Chain Classes of Homeomorphisms of Compact Metric Spaces

The strong chain recurrent points and strong chain transitive sets of a homeomorphism f on a compact metric space X, first introduced by Easton [4]. He note that, if chain recurrent set of f is all of X, then strong chain recurrent set and strong chain transitive set of f are useful. Here, we obtain some results about strong chain recurrency. In particular, we show that an isolated strong chain class S of generic homeomorphism f has a generic continuation Sg , where g close to f, in C ○-topology. Moreover, we have the generic persistence of Lipschitz ergodicity at f near S.

An Algorithmic Approach to Chain Recurrence

In this paper we give a new definition of the chain recurrent set of a continuous map using finite spatial discretizations. This approach allows for an algorithmic construction of isolating blocks for the components of Morse decompositions which approximate the chain recurrent set arbitrarily closely as well as discrete approximations of Conley’s Lyapunov function. This is a natural framework in which to develop computational techniques for the analysis of qualitative dynamics including rigorous computer-assisted proofs.

Dynamics of topologically generic homeomorphisms

Introduction Attractors and chain recurrence Periodic decompositions and adding machines Semicontinuity and homogeneity Crushing arguments Topological horseshoes Generic homeomorphisms Almost equicontinuity Cantor sets The circle Crushing the chain recurrent set Generic homeomorphisms on manifolds Generic mappings on manifolds Bibliography Index.

Chain recurrence in flows on the Klein bottle

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

Chain recurrence and discretisation

Upper and lower semicontinuity results for the chain recurrent set are shown to remain valid in numerical dynamics with constant stepsizes. It is also pointed out that the chain recurrent set contains numerical ω–limit sets for discretisations with a variable stepsize sequence approaching zero.

Almost periodicity, chain recurrence and chaos

In this paper we consider a continuous mapf: X→X, whereX is a compact metric space. The existence of chaotic sets off is discussed. For the special caseX=[0,1], we prove thatf has a positive topological entropy iff it has an uncountable chaotic set in which each point is almost periodic, and iff it has an uncountable chaotic set in which each point is chain recurrent. As an application, a uniform proof for some known results will be given.