Erdős measures on the Euclidean space and on the group of $A$-adic integers

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 $P(\xi_t=j)=p_j$, $j\in\mathbb Z^n$, $0<p_0<1$, $\sum_j p_j=1$. 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.