Les suites de Rudin-Shapiro ont des propriétés extrémales en analyse harmonique. En remarquant qu’une telle suite est reconnaissable par un automate fini, nous en décrivons explicitement le spectre (type spectral maximal, multiplicité spectrale fonction multiplicité). Nous établissons par exemple, que la suite de Rudin-Shapiro généralisée à l’ordre $q$ contient dans son spectre une composante de Lebesgue, de multiplicité $q\phi \left(q\right)$.

Any sequence $x={\left(x_k\right)}_{k=1}^{\infty }$ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let $H_n\left(x\right)$ be the height of the tree generated by $x_1,\cdots ,x_n.$ Obviously$$\frac{logn}{log2}-1\le H_n\left(x\right)\le n-1.$$If the sequences $x$ are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists $c$ > $0$ such that $H_n\left(x\right)\sim clogn$ as $n\to \infty$ for almost all sequences $x$. Recently...

Let $\sigma$ be a unimodular Pisot substitution over a $d$ letter alphabet and let $X_1,...,X_d$ be the associated Rauzy fractals. In the present paper we want to investigate the boundaries $\partial X_i$ ($1\le i\le d$) of these fractals. To this matter we define a certain graph, the so-called contact graph$𝒞$ of $\sigma$. If $\sigma$ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries $\partial X_1,...,\partial X_d$. From this graph...

For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_i\in 0,1$. We prove that for any x ∈ (0,1), (x) contains a sequence ${{\beta }_k}_{k\ge 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\overline{\left(x\right)}$ are nowhere dense.

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.