The Chowla and the Sarnak conjectures from ergodic theory point of view
El Houcein El Abdalaoui Joanna Kułaga-Przymus Mariusz Lemańczyk Thierry de la Rue
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 2899-2944 doi: 10.3934/dcds.2017125

We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in $\{-1,0,1\}^{{\mathbb{N}^*}}$, and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.

keywords: Chowla conjecture Sarnak conjecture Möbius orthogonality ergodic theory theory of joinings
An introduction to joinings in ergodic theory
Thierry de la Rue
Discrete & Continuous Dynamical Systems - A 2006, 15(1): 121-142 doi: 10.3934/dcds.2006.15.121
Since their introduction by Furstenberg [3], joinings have proved a very powerful tool in ergodic theory. We present here some aspects of the use of joinings in the study of measurable dynamical systems, emphasizing
  • the links between the existence of a non trivial common factor and the existence of a joining which is not the product measure,
  • how joinings can be employed to provide elegant proofs of classical results,
  • how joinings are involved in important questions of ergodic theory, such as pointwise convergence or Rohlin's multiple mixing problem.
keywords: minimal self-joinings disjointness Joinings weak disjointness 2-fold and 3-fold mixing.
Dynamics of $\lambda$-continued fractions and $\beta$-shifts
Élise Janvresse Benoît Rittaud Thierry de la Rue
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1477-1498 doi: 10.3934/dcds.2013.33.1477
For a real number $0<\lambda<2$, we introduce a transformation $T_\lambda$ naturally associated to expansion in $\lambda$-continued fraction, for which we also give a geometrical interpretation. The symbolic coding of the orbits of $T_\lambda$ provides an algorithm to expand any positive real number in $\lambda$-continued fraction. We prove the conjugacy between $T_\lambda$ and some $\beta$-shift, $\beta>1$. Some properties of the map $\lambda\mapsto\beta(\lambda)$ are established: It is increasing and continuous from $]0, 2[$ onto $]1,\infty[$ but non-analytic.
keywords: symbolic dynamics homographic functions maps of the interval. Continued fraction beta-expansion

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