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### Open Access Journals

DCDS

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

DCDS

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.

DCDS

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.

## Year of publication

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