A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity

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