Infinite products of $2 imes2$ matrices and the Gibbs properties of Bernoulli convolutionsReport as inadecuate




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1 LATP - Laboratoire d-Analyse, Topologie, Probabilités

Abstract : 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 $\displaystyle{A 1\dots A nV\over\vert\vert A 1\dots A nV\vert\vert}$ to converge uniformly. In application we prove that the Bernoulli convolutions related to the numeration in Pisot quadratic bases are weak Gibbs.

Keywords : Infinite products of matrices weak Gibbs measures Bernoulli convolutions Pisot numbers $\beta$-numeration





Author: Eric Olivier - Alain Thomas -

Source: https://hal.archives-ouvertes.fr/



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