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On the ω-limit sets of tent maps

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Barwell A., Davies G., Good C.

Fundamenta Mathematicae

2012

tom 217

nr 1

35-54

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 = x₀,x₁,...,xₙ = y⟩ such that $d(f(x_i),x_{i+1})< δ$ for 0 ≤ i< n. 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 tent maps with shadowing, the ω-limit sets are precisely those sets having internal chain transitivity.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Barwell A.

1 Davies G.

1 Good C.

1 2012

Wyniki wyszukiwania

On the ω-limit sets of tent maps