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1

A characterization of ω-limit sets for piecewise monotone maps of the interval

100%

Barwell A.

Fundamenta Mathematicae

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2010

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tom 207

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nr 2

161-174

EN

For a piecewise monotone map f on a compact interval I, we characterize the ω-limit sets that are bounded away from the post-critical points of f. If the pre-critical points of f are dense, for example when f is locally eventually onto, and Λ ⊂ I is closed, invariant and contains no post-critical point, then Λ is the ω-limit set of a point in I if and only if Λ is internally chain transitive in the sense of Hirsch, Smith and Zhao; the proof relies upon symbolic dynamics. By identifying points of ω-limit sets via their limit-itineraries, we offer simple examples which show that internal chain transitivity does not characterize ω-limit sets for interval maps in general.

2

On the ω-limit sets of tent maps

51%

Barwell A., Davies G., Good C.

Fundamenta Mathematicae

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2012

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tom 217

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nr 1

35-54

EN

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.

3

Shadowing and expansivity in subspaces

51%

Barwell A., Good C., Oprocha P.

Fundamenta Mathematicae

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2012

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tom 219

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nr 3

223-243

EN

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Barwell A.

2 Good C.

1 Davies G.

1 Oprocha P.

2 2012

1 2010

A characterization of ω-limit sets for piecewise monotone maps of the interval

On the ω-limit sets of tent maps

Shadowing and expansivity in subspaces