Chain transitivity and uniform persistence

Abstract. For a continuous map f on a compact metric space (X, d), a set D ⊂ X is internally chain transitive if for every x, y ∈ D and every δ > 0 there is a sequence of points x = x 0 , x 1 , . . . , x n = y such that d(f (x i ), x i+1 ) < δ for 0 ≤ i < n. It is known that every ω-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed set D ⊂ X is internally chain transitive if and only if D is an ω-limit set for some point in X, and that the same is also true for the full tent map T 2 : [0, 1] → [0, 1]. In this paper, we prove that for tent maps with periodic critical point every closed, internally chain transitive set is necessarily an ω-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 ω-limit set. Together, these results lead us to conjecture that for maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

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On the ω-limit sets of tent maps

Barwell

Davies

Good

2012

Fund. Math.

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Chain transitivity and uniform persistence

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On the ω-limit sets of tent maps

On the ω-limit sets of tent maps

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