Recurrence of states in a function of a Markov chain

Question

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence status of $f(s)$ in $f(X)$? The motivation behind my question is the simple symmetric random walk in $\Bbb Z$ as $X$ and $f(X)=|X|$. Unless I'm mistaken, both chains are irreducible and recurrent.

Answer 1

That $s$ is recurrent means that $X$ visits $s$ with full probability. Thus, $f(X)$ visits $f(s)$ with full probability, that is, $f(s)$ is recurrent for $f(X)$.