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 $f^{n}(U)\cap B_{\delta}(V)\neq\varnothing.$ We introduce $\delta$-topological transitivity, $\delta$-topological mixing, and a uniform-from-below version of $\delta$-mixing, and prove $\mathrm{UFB\mbox{-}}\delta\text{-TM} \;\Rightarrow\; \delta\text{-TM} \;\Rightarrow\; \delta\text{-TT}.$ 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 $\delta>0$.