Hypercyclic and topologically transitive semigroups of composition operators (revised and extended version)

Abstract

Let X be a separable locally compact metric space, let E≠{0} be a normed space, and let G be a semigroup of continuous self-maps of X. Every φ∈ G induces the composition operator C_φf=f∘φ on the space C(X,E) of continuous functions endowed with the compact-open topology. We characterize the topological transitivity of the semigroup Ĝ={C_φ:φ∈ G} by a condition that combines the run-away property of G with an approximate form of injectivity, and we show that, when E is separable, Ĝ is topologically transitive if and only if it is hypercyclic. In particular, if every element of G is one-to-one and G is run-away, then Ĝ is topologically transitive, and hypercyclic when E is separable. For a single map φ we obtain: C_φ is hypercyclic ⟺ C_φ is topologically transitive ⟺ φ is one-to-one and run-away, and C_φ is mixing ⟺ φ is one-to-one and strongly run-away. If, moreover, φ is an open map (which is automatic on manifolds by invariance of domain), then C_φ is Devaney chaotic. On an open interval all of these properties are equivalent to φ being one-to-one without fixed points.

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BibTeX RIS

BibTeX

@misc{javaheri2026hypercyclic,
  title = {{Hypercyclic and topologically transitive semigroups of composition operators (revised and extended version)}},
  author = {Mohammad Javaheri},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08320},
  eprint = {2611.08320},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08320},
}

RIS

TY  - UNPB
TI  - Hypercyclic and topologically transitive semigroups of composition operators (revised and extended version)
AU  - Mohammad Javaheri
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08320
UR  - https://arxiv.org/abs/2611.08320
AB  - Let X be a separable locally compact metric space, let E≠{0} be a normed space, and let G be a semigroup of continuous self-maps of X. Every φ∈ G induces the composition operator C_φf=f∘φ on the space C(X,E) of continuous functions endowed with the compact-open topology. We characterize the topological transitivity of the semigroup Ĝ={C_φ:φ∈ G} by a condition that combines the run-away property of G with an approximate form of injectivity, and we show that, when E is separable, Ĝ is topologically transitive if and only if it is hypercyclic. In particular, if every element of G is one-to-one and G is run-away, then Ĝ is topologically transitive, and hypercyclic when E is separable. For a single map φ we obtain: C_φ is hypercyclic ⟺ C_φ is topologically transitive ⟺ φ is one-to-one and run-away, and C_φ is mixing ⟺ φ is one-to-one and strongly run-away. If, moreover, φ is an open map (which is automatic on manifolds by invariance of domain), then C_φ is Devaney chaotic. On an open interval all of these properties are equivalent to φ being one-to-one without fixed points.
ER  -

Abstract

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