The generalized recurrent set and strong chain recurrence

Abstract

Fathi and Pageault have recently shown a connection between Auslander's generalized recurrent set $GR(f)$ and Easton's strong chain recurrent set. We study $GR(f)$ 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 $GR(f^k)=GR(f)$ for any $k>0$, and give a characterization of maps for which the generalized recurrent set is different from the ordinary chain recurrent set.Comment: Revised version incorporates referee's comment

Generalized recurrence and the nonwandering set for products

“…The interesting dynamics occurs on the nonwandering set, so in order to understand the relationship between the dynamics of a product map and the dynamics of the original maps, we need to understand the nonwandering set; we give necessary and sufficient conditions (Theorem 3.11) for a point x ∈ X to be product nonwandering, that is, for (x, y) to be nonwandering for f × g for any g and any nonwandering point y ∈ Y . Auslander's generalized recurrent set GR(f ) (defined originally for flows (see [4]), and extended to maps (see [1,3])) is a larger and in many ways more dynamically natural set, particularly for understanding Lyapunov functions; see [9] and the references in [17]. We show that GR(f × g) = GR(f ) × GR(g) (Theorem 3.1).…”

“…[17, Theorem 3.3]). A point x is in M(f ) if and only if for any closed neighborhood D of the diagonal in X × X, there exist a closed symmetric neighborhood N of the diagonal and an integer n > 0 such that N 3 n ⊂ D and there is an (N , f )-chain of length n from x to itself.…”

The Generalized Recurrent Set, Explosions and Lyapunov Functions

“…The set of strong chain recurrent points is denoted by SCR d (f ). In general, strong chain recurrence depends on the choice of the metric; see for example [21][Example 3.1] and [19][Example 2 .6]. A way to eliminate the dependence on the metric in SCR d (f ) is taking the intersection over all metrics.…”

“…where N W(f ) denotes the non-wandering set of f , and all inclusions can be strict. We refer to [19][Example 2.9] for an exhaustive treatment of these inclusions. The dynamical relevance of the generalized recurrent set relies on its relations with continuous Lyapunov functions.…”

A Conley-type decomposition of the strong chain recurrent set

“…The properties of the relation SP(X) are illustrated in the next proposition, see also [17] (Definition 3.2) and [18] (Definition 2.3 and Theorems 2.1, 2.2, 2.3). We refer to [5] (Page 36, Section 6) for the analogous result in the chain recurrent case.…”