A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913047388356608 |
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| author | Alshammari, Hadi Obaid Benchiheb, Otmane Chiotis, Dimitrios |
| author_facet | Alshammari, Hadi Obaid Benchiheb, Otmane Chiotis, Dimitrios |
| contents | 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 $δ$-topological transitivity, $δ$-topological mixing,
and a uniform-from-below version of $δ$-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
$δ$-Hypercyclicity Criterion, prove that it implies
$δ$-hypercyclicity, and show that the classical Hypercyclicity Criterion
implies the $δ$-criterion for every $δ>0$. We further show that
this criterion yields eventual $δ$-mixing along the underlying sequence.
Finally, we discuss weighted backward shifts, derive sufficient conditions for
$δ$-topological mixing, and show that $λB$ satisfies the
$δ$-Hypercyclicity Criterion for every $δ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity Alshammari, Hadi Obaid Benchiheb, Otmane Chiotis, Dimitrios Functional Analysis Dynamical Systems 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 $δ$-topological transitivity, $δ$-topological mixing, and a uniform-from-below version of $δ$-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 $δ$-Hypercyclicity Criterion, prove that it implies $δ$-hypercyclicity, and show that the classical Hypercyclicity Criterion implies the $δ$-criterion for every $δ>0$. We further show that this criterion yields eventual $δ$-mixing along the underlying sequence. Finally, we discuss weighted backward shifts, derive sufficient conditions for $δ$-topological mixing, and show that $λB$ satisfies the $δ$-Hypercyclicity Criterion for every $δ>0$. |
| title | A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity |
| topic | Functional Analysis Dynamical Systems |
| url | https://arxiv.org/abs/2604.18142 |