A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity

TL;DR

This paper introduces metric-based versions of transitivity, mixing, and hypercyclicity for continuous maps, providing new criteria and implications, especially in linear spaces and for weighted shifts.

Contribution

It develops a comprehensive metric framework for approximate dynamical properties and establishes new criteria linking classical and metric notions.

Findings

Proves implications among various $oldsymbol{ ext{delta}}$-properties.

Introduces a $oldsymbol{ ext{delta}}$-Hypercyclicity Criterion implying $oldsymbol{ ext{delta}}$-hypercyclicity.

Shows weighted backward shifts satisfy the $oldsymbol{ ext{delta}}$-Hypercyclicity Criterion for all $oldsymbol{ ext{delta}} > 0$.

Abstract

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 $\delta>0$.