PreprintarXiv licencearXiv
Some observations on transitivity of Lipschitz operators
Abstract
The universal property of the Lipschitz-free spaces allows for the linearisation of a base point preserving Lipschitz map f: M→ M to a bounded linear operator Tf: ℱ(M) → ℱ(M). While it is known that the operator Tf is weakly mixing whenever f has this property, it is open whether a similar inheritance result for topological transitivity holds. Motivated by this question we investigate properties of Tf under the assumption that f is topologically transitive. In particular we prove Kitai's theorem for Lipschitz operators, i.e. we show that every connected component of the complex variant of Tf intersects the unit circle if f is topologically transitive.
Related
- Hypercyclicity and Lipschitz-free operators
- Littlewood-Paley and Carleson measure characterizations of Lipschitz spaces adapted to Schrödinger operators
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- Hypercyclic Bergman-Toeplitz operators with some harmonic symbols
- Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces
BibTeX
@misc{bargetz2026some,
title = {{Some observations on transitivity of Lipschitz operators}},
author = {Christian Bargetz and Leon Kügler},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07921},
eprint = {2609.07921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07921},
}
RIS
TY - UNPB
TI - Some observations on transitivity of Lipschitz operators
AU - Christian Bargetz
AU - Leon Kügler
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.07921
UR - https://arxiv.org/abs/2609.07921
AB - The universal property of the Lipschitz-free spaces allows for the linearisation of a base point preserving Lipschitz map f: M→ M to a bounded linear operator T_f: ℱ(M) → ℱ(M). While it is known that the operator T_f is weakly mixing whenever f has this property, it is open whether a similar inheritance result for topological transitivity holds. Motivated by this question we investigate properties of T_f under the assumption that f is topologically transitive. In particular we prove Kitai's theorem for Lipschitz operators, i.e. we show that every connected component of the complex variant of T_f intersects the unit circle if f is topologically transitive.
ER -