arXiv subjects
Publications and source records attributed to Fabien Durand.
We give necessary and sufficient conditions to have measurable and continuous eigenfunctions for linearly recurrent Cantor dynamical systems. We also construct explicitly an example of linearly recurrent system with nontrivial Kronecker factor and a trivial maximal equicontinuous factor.
In this article we generalize Cobham theorem to a large class of substitutions including non primitive and non constant length substitutions.
Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and V recognizable, then E is a finite union of arithmetical progressions.
We say $x \in \{0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any $a\in \{0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.
In this paper we prove that if two self-similar tiling systems, with respective stretching factors $λ_1$ and $λ_2$, have a common factor which is a non periodic tiling system, then $λ_1$ and $λ_2$ are multiplicatively dependent.
We give a new proof of the Cobham's first theorem using ideas from symbolic dynamics and of the Cobham-Semenov theorem (in the primitive case) using ideas from tiling dynamics.