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A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled

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Abstract

We give a simple proof that any dendrite map for which the set of endpoints is closed and countable fulfilled Sarnak Möbius disjointness. We further notice that the Smital-Ruelle property can be extended to the class of dendrites with closed and countable endpoints. Our proof use only Dirichlet theorem.

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Notes

  1. This result is from unpublished notes of Veech [39]. The notes was sent to the first author by W. Veech. In his letter he indicated that there is only four persons in the world who has a copy of this notes ( W. Veech wrote “Peter, Jon Fickenscher (a PHD student of mine in Princeton), one non-ergodicist friend in Princeton (who is Godfather to my children and now you", see the entire letter of Veech in [14].) )

  2. \(X^X\) is equipped with the pointwise convergence. This closure is called the enveloping semigroup of Ellis.

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Acknowledgements

This paper is a part of the Master thesis by the second author. The authors would like to thanks G. Askri, K. Dajani, H. Marzougui, M. Nerurkar, O. Sarig, and X-D. Ye for their comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, University of Rouen Normandy, LMRS UMR 60 85 CNRS, Saint Etienne du Rouvray, France

    el Houcein el Abdalaoui

  2. Ecole normale supérieure Paris-Saclay, 61, Avenue du Président Wilson, 94230, Cachan, France

    Joseph Devianne

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  1. el Houcein el Abdalaoui
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el Abdalaoui, e.H., Devianne, J. A Large Class of Dendrite Maps for Which Möbius Disjointness Property of Sarnak is Fulfilled. Qual. Theory Dyn. Syst. 23, 83 (2024). https://doi.org/10.1007/s12346-023-00941-6

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