%0 Unpublished work
%T How to prove that some Bernoulli convolution has the weak Gibbs property
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%A Olivier, Éric
%A Thomas, Alain
%Z 27 pages
%8 2010-07-29
%D 2010
%Z 1006.3616
%K Infinite products of nonnegative matrices
%K Gibbs properties
%K multifractal analysis of mesures
%K Bernoulli convolutions
%Z MSC 11A67, 15A48, 28A78, 28A80
%Z Mathematics [math]/General Mathematics [math.GM]Preprints, Working Papers, ...
%X In this paper we give an example of uniform convergence of the sequence of column vectors $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$, $A_i\in\{A,B,C\}$, $A,B,C$ being some $(0,1)$-matrices of order $7$ with much null entries, and $V$ a fixed positive column vector. These matrices come from the study of the Bernoulli convolution in the base $\beta>1$ such that $\beta^3=2\beta^2-\beta+1$, that is, the (continuous singular) probability distribution of the random variable $\displaystyle(\beta-1)\sum_{n=1}^\infty{\omega_n\over\beta^n}$ when the independent random variables $\omega_n$ take the values $0$ and $1$ with probability $\displaystyle{1\over2}$. In the last section we deduce, from the uniform convergence of $\displaystyle{A_1\dots A_nV\over\left\Vert A_1\dots A_nV\right\Vert}$, the Gibbs and the multifractal properties of this measure.
%G English
%2 https://hal.science/hal-00493068v2/document
%2 https://hal.science/hal-00493068v2/file/ddmatrices2.pdf
%L hal-00493068
%U https://hal.science/hal-00493068
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ I2M