On the omega-limit sets of tent maps

TL;DR

This paper investigates the relationship between internally chain transitive sets and omega-limit sets in tent maps, proving equivalence in some cases and providing counterexamples in others, advancing understanding of dynamical behavior.

Contribution

It proves that for tent maps with a periodic critical point, all internally chain transitive sets are omega-limit sets, and identifies cases where this does not hold.

Findings

For tent maps with periodic critical points, internally chain transitive sets are omega-limit sets.

Existence of non-omega-limit, internally chain transitive sets in tent maps with non-recurrent critical points.

Conjecture that shadowing maps have omega-limit sets exactly characterized by internal chain transitivity.

Abstract

For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prove that for tent maps whose critical point c=1/2 is periodic, every closed, internally chain transitive set is necessarily an omega-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an omega-limit set. Together, these results lead us to conjecture that for those tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are precisely those sets having internal chain transitivity.