Dynamics on sensitive and equicontinuous functions
KeywordsSensitivity, sensitive pairs, sensitive functions, equicontinuous functions, weak mixing
AbstractThe notions of sensitive and equicontinuous functions under semigroup action are introduced and intensively studied. We show that a transitive system is sensitive if and only if it has a sensitive pair if and only if it has a sensitive function. While there exists a minimal non-weakly mixing system such that every non-constant continuous function is sensitive, and a topological dynamical system is weakly mixing if and only if it is sensitive consistently with respect to (at least) any two non-constant continuous functions. We also get a dichotomy result for minimal systems -- every continuous function is either sensitive or equicontinuous.
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