Geodesic
Erdős measures on the Euclidean space and on the group of $A$-adic integers
Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 38-45 
Z. I. Bezhaeva; V. L. Kulikov; E. F. Olekhova; V. I. Oseledets. Erdős measures on the Euclidean space and on the group of $A$-adic integers. Informatics and Automation, Order and chaos in dynamical systems, Tome 297 (2017), pp. 38-45. http://geodesic.mathdoc.fr/item/TRSPY_2017_297_a1/
@article{TRSPY_2017_297_a1,
     author = {Z. I. Bezhaeva and V. L. Kulikov and E. F. Olekhova and V. I. Oseledets},
     title = {Erd\H{o}s measures on the {Euclidean} space and on the group of $A$-adic integers},
     journal = {Informatics and Automation},
     pages = {38--45},
     year = {2017},
     volume = {297},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2017_297_a1/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $A\in M_n(\mathbb Z)$ be a matrix with eigenvalues greater than $1$ in absolute value. The $\mathbb Z^n$-valued random variables $\xi_t$, $t\in\mathbb Z$, are i.i.d., and $P(\xi_t=j)=p_j$, $j\in\mathbb Z^n$, $0, $\sum_j p_j=1$. We study the properties of the distributions of the $\mathbb R^n$-valued random variable $\zeta_1=\sum_{t=1}^\infty A^{-t}\xi_t$ and of the random variable $\zeta=\sum_{t=0}^\infty A^t\xi_{-t}$ taking integer $A$-adic values. We obtain a necessary and sufficient condition for the absolute continuity of these distributions. We define an invariant Erdős measure on the compact abelian group of $A$-adic integers. We also define an $A$-invariant Erdős measure on the $n$-dimensional torus. We show the connection between these invariant measures and functions of countable stationary Markov chains. In the case when $|\{j\colon p_j\ne 0\}|<\infty$, we establish the relation between these invariant measures and finite stationary Markov chains.

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