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\({\mathcal F}\)-limit points in dynamical systems defined on the interval. (English) Zbl 1338.37050

Summary: Given a free ultrafilter \(p\) on \(\mathbb{N}\) we say that \(x\in [0, 1]\) is the \(p\)-limit point of a sequence \((x_n)_ {n\in\mathbb{N}}\subset [0, 1]\) (in symbols, \(x = p -\lim_{n\in\mathbb{N}} x_n)\) if for every neighbourhood \(V\) of \(x\), \(\{n\in\mathbb{N} : x_n\in V\}\in p\). For a function \(f: [0, 1] \to [0, 1]\) the function \(f^p : [0, 1] \to [0, 1]\) is defined by \(f^p (x) = p -\lim_{n\in\mathbb{N}} f^n(x)\) for each \(x\in [0, 1]\). This map is rarely continuous. In this note we study properties which are equivalent to the continuity of \(f^p\). For a filter \(\mathcal{F}\) we also define the \(\omega^{\mathcal{F}}\)-limit set of \(f\) at \(x\). We consider a question about continuity of the multivalued map \(x\to\omega_f^{\mathcal{F}}(x)\). We point out some connections between the Baire class of f p and tame dynamical systems, and give some open problems.

MSC:

37E05 Dynamical systems involving maps of the interval
03E15 Descriptive set theory
26A03 Foundations: limits and generalizations, elementary topology of the line
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
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