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Setwise quasicontinuity and \(\pi\)-related topologies. (English) Zbl 1011.54033

The authors introduce the notion of a setwise quasicontinuous collection of functions and show that the set of iterates \(\{ 1_X,f,f\circ f,\dots \} \) of a self-map \(f:X \to X\) is setwise quasicontinuous if and only if the topology can be extended to a \(\pi\)-related one, so that each iterate is continuous from the new space to the new space. Moreover they present a quasicontinuous function on the unit interval which is discontinuous on a dense subset and show that conjugacies of dynamical systems via quasicontinuous bijections preserve much of the desired structure of the systems.

MSC:

54H20 Topological dynamics (MSC2010)
37B99 Topological dynamics
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