= Ergodic invariant probability on a countable state space has finite cyclic support
Let $T$ be an <ergodic transformation> preserving a probability measure $\mu$ on a <countable set>. Some state $x$ has $\mu(\{x\})>0$. Along its forward orbit,
$$
\mu(\{T^{n+1}x\})
=\mu(T^{-1}\{T^{n+1}x\})
\geq\mu(\{T^nx\}).
$$
An infinite sequence of distinct states with masses bounded below by $\mu(\{x\})$ is impossible, so the orbit eventually reaches a finite cycle $Y$. Since $Y\subseteq T^{-1}Y$ and invariance gives $\mu(T^{-1}Y)=\mu(Y)>0$, the difference is null. The cycle is invariant modulo null sets, and ergodicity therefore gives $\mu(Y)=1$.
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