Chain recurrence

The most general of the properties expressing "repetition of motions" considered in topological dynamics. In the basic case of a topological flow $\{S_t\}$ on a compact metric space $W$ with metric $\rho$, a point $w\in W$ has the property of chain recurrence if for every $\epsilon,T>0$ there is an $\epsilon$-trajectory starting in $w$ and again returning to $w$ after a time $T_\epsilon>T$. An $\epsilon$-trajectory is a parametrized (possibly discontinuous) curve $w(t)$, $0\leq t\leq T_\epsilon$, such that $\rho(S_\tau w(t),w(t+\tau))<\epsilon$ for $0\leq\tau\leq1$, $0\leq t\leq T_\epsilon-1$ ("a finite segment of an -trajectory is close to a segment of the actual trajectory of a dynamical system28Dxx34Cxx37-XX37-XX54H2054H20trajectory"). There is also a definition of chain recurrence for a more general case [1]. If $W$ is a closed manifold, then chain recurrence is the same as the property of "weak non-wandering" (see [3]), which reflects more directly the influence of small perturbations (in the topological sense) of the system on the behaviour of its trajectories. Outside the set of points with the property of chain recurrence the behaviour of the system resembles that of a gradient dynamical system (see [1], [2]).

References

[1]	C. Conley, "Isolated invariant sets and the Morse index" , Amer. Math. Soc. (1978)
[2]	M. Shub, "Global stability of dynamical systems" , Springer (1987) (Translated from French)
[3]	A.N. Sharkovskii, V.A. Dobrynskii, , Dynamical systems and problems of stability of solutions of differential equations , Kiev (1973) pp. 125–174 (In Russian)

Comments

Concerning stability properties of chain-recurrent sets, consult also [a1]. The notion of an $\epsilon$-trajectory (and the related notion of $\epsilon$-shadowing) is important in the study of hyperbolic sets (cf. Hyperbolic set) in dynamical systems; see Chapt. 2 in [a2].

References

[a1]	L. Block, J.E. Franke, "The chain recurrent set, attractors, and explosions" Ergodic Theory and Dynamical Systems , 5 (1985) pp. 321–327
[a2]	R. Bowen, "On Axiom $A$ diffeomorphisms" , Amer. Math. Soc. (1978)