I have a problem on my ergodic theory course: Let $X$ be a symbolic compact of infinite words like $x_1x_2x_3...$. Each $x_i$ equals to one of the symbols from the alphabet $d=\{1, 2, 3, ..., d\}$. There are also given transfer probabilities $p_i$ to transfer from $x$ to $y=ix$, i.e. if $x = x_1x_2x_3...$ then $y=ix_1x_2x_3...$ . As usual $p_1 + p_2 + ... + p_d = 1$. The question is to find all the invariant measures.
There is also a question about $P$ - transfer (markov) operator of this chain. The question is - is it true that $P^nf$ uniformly converge in $C(X)$ and what is its limit? Any help would be appreciated.