The generalized recurrent set and strong chain recurrence

TL;DR

This paper explores the generalized recurrent set in dynamical systems, establishing metric-independent definitions, invariance under iteration, and characterizing maps with distinct generalized and chain recurrent sets.

Contribution

It provides new metric-free definitions of the generalized recurrent set, proves its invariance under iteration, and characterizes when it differs from the chain recurrent set.

Findings

$GR(f^k)=GR(f)$ for all $k>0$

Equivalent metric-free definitions of $GR(f)$

Characterization of maps with different $GR(f)$ and chain recurrent set

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.