How to prove that some Bernoulli convolution has the weak Gibbs property

Details How to prove that some Bernoulli convolution has the weak Gibbs property How to prove that some Bernoulli convolution has the weak Gibbs property

2010-06-18

arxiv.org/abs/1006.3616v2

Mathematics

Dynamical Systems

27 pages

Scientific paper

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.

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