On the structure of invariant measures related to noncommutative random products

Let $G=SL(R,n)$ be the group of mappings of the real projective space $P^{n-1}$ onto itself. There is introduced the notion of a boundary measure $\nu$ on $P^{n-1}$ for a probability measure $\mu$ on $G$, and its relation to the unique invariant measure on $P^{n-1}$ with respect to the operator $\pi(x,A)=\mu\{g\in G:gx\in A\}$ is found. It is established that the Markov chain generated by the transition probability $\pi(x,A)$ and the invariant boundary measure $\nu$ is a factor-automorphism of an automorphism of a certain Bernoulli space. A limit theorem for random mappings of a segment of the line into itself is proved.
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