A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity

Abstract

We study metric versions of transitivity, mixing, and hypercyclicity for continuous maps, based on intersections of the form \( f^{n}(U)\cap B_δ(V)\neq\varnothing. \) We introduce $\hat{I}´$-topological transitivity, $\hat{I}´$-topological mixing, and a uniform-from-below version of $\hat{I}´$-mixing, and prove \( \mathrm{UFB\mbox{-}}δ\text{-TM} \;\Rightarrow\; δ\text{-TM} \;\Rightarrow\; δ\text{-TT}. \) In the linear setting of separable F-spaces, we formulate a $\hat{I}´$-Hypercyclicity Criterion, prove that it implies $\hat{I}´$-hypercyclicity, and show that the classical Hypercyclicity Criterion implies the $\hat{I}´$-criterion for every $\hat{I}´>0$. We further show that this criterion yields eventual $\hat{I}´$-mixing along the underlying sequence. Finally, we discuss weighted backward shifts, derive sufficient conditions for $\hat{I}´$-topological mixing, and show that $\hat{I}»B$ satisfies the $\hat{I}´$-Hypercyclicity Criterion for every $\hat{I}´>0$.