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F-limit points in dynamical systems defined on the interval

Piotr Szuca — 2013

Open Mathematics

Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider...

The Covering Principle for Darboux Baire 1 functions

Piotr Szuca — 2007

Fundamenta Mathematicae

We show that the Covering Principle known for continuous maps of the real line also holds for functions whose graph is a connected G δ subset of the plane. As an application we find an example of an approximately continuous (hence Darboux Baire 1) function f: [0,1] → [0,1] such that any closed subset of [0,1] can be translated so as to become an ω-limit set of f. This solves a problem posed by Bruckner, Ceder and Pearson [Real Anal. Exchange 15 (1989/90)].

Sharkovskiĭ's theorem holds for some discontinuous functions

Piotr Szuca — 2003

Fundamenta Mathematicae

We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected G δ graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected G δ graph.

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz NatkaniecPiotr Szuca — 2010

Colloquium Mathematicae

We prove that if f: → is Darboux and has a point of prime period different from 2 i , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

When does the Katětov order imply that one ideal extends the other?

Paweł BarbarskiRafał FilipówNikodem MrożekPiotr Szuca — 2013

Colloquium Mathematicae

We consider the Katětov order between ideals of subsets of natural numbers (" K ") and its stronger variant-containing an isomorphic ideal ("⊑ "). In particular, we are interested in ideals for which K for every ideal . We find examples of ideals with this property and show how this property can be used to reformulate some problems known from the literature in terms of the Katětov order instead of the order "⊑ " (and vice versa).

Ideal version of Ramsey's theorem

Rafał FilipówNikodem MrożekIreneusz RecławPiotr Szuca — 2011

Czechoslovak Mathematical Journal

We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....

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