Möbius disjointness for models of an ergodic system and beyond

Autor: Joanna Kułaga-Przymus, El Houcein El Abdalaoui, Mariusz Lemańczyk, Thierry de la Rue
Přispěvatelé: Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Faculty of Mathematics and Computer Science, Nicolaus Copernicus University [Toruń], Research supported by the special program in the framework of the Jean Morlet semester'Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combina-torics', and by Narodowe Centrum Nauki grant UMO-2014/15/B/ST1/03736., Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
Rok vydání: 2018
Předmět:
Zdroj: Israël Journal of Mathematics
Israël Journal of Mathematics, Hebrew University Magnes Press, 2018, 228 (2), pp.707-751. ⟨10.1007/s11856-018-1784-z⟩
Israel Journal of Mathematics
Israel Journal of Mathematics, 2018, 228 (2), pp.707-751. ⟨10.1007/s11856-018-1784-z⟩
ISSN: 1565-8511
0021-2172
DOI: 10.1007/s11856-018-1784-z
Popis: Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},\kappa,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are M\"obius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbol{\mu}$ in all zero entropy systems, in particular, it makes $\boldsymbol{\mu}$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.
Comment: 35 pages
Databáze: OpenAIRE
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