arXiv subjects

Eric Olivier

Publications and source records attributed to Eric Olivier.

3 recordsLinked to original sources

arXiv2006

Infinite products of $2\times2$ matrices and the Gibbs properties of Bernoulli convolutions

We consider the infinite sequences $(A\_n)\_{n\in\NN}$ of $2\times2$ matrices with nonnegative entries, where the $A\_n$ are taken in a finite set of matrices. Given a vector $V=\pmatrix{v\_1\cr v\_2}$ with $v\_1,v\_2>0$, we give a necessary and sufficient condition for $\displaystyle{A\_1... A\_nV\over|| A\_1... A\_nV||}$ to converge uniformly. In application we prove that the Bernoulli convolutions related to the numeration in Pisot quadratic bases are weak Gibbs.

math.NT↗

arXiv2006

Weak Gibbs property and system of numeration

We study the selfsimilarity and the Gibbs properties of several measures defined on the product space $Ω\_r:=\{0,1,...,\break r-1\}^{\mathbb N}$. This space can be identified with the interval $[0,1]$ by means of the numeration in base $r$. The last section is devoted to the Bernoulli convolution in base $β={1+\sqrt5\over2}$, called the Erd\H os measure, and its analogue in base $-β=-{1+\sqrt5\over2}$, that we study by means of a suitable system of numeration.

math.NT↗

arXiv2004

On the Gibbs properties of Bernoulli convolutions related to $β$-numeration in multinacci bases

We consider infinitely convolved Bernoulli measures (or simply Bernoulli convolutions) related to the $β$-numeration. A matrix decomposition of these measures is obtained in the case when $β$ is a PV number. We also determine their Gibbs properties for $β$ being a multinacci number, which makes the multifractal analysis of the corresponding Bernoulli convolution possible.

math.NT↗