Let $\{ X_n(\omega,x)\}_{n \ge 0}$ be a Markov chain with and underlying probability space $(\Omega,\Sigma,\mathbb{P})$ and state space $X= \mathbb{S}^1$. Suppose this markov chain admits unique ergodic measure which is full supported.

I would like to estimate $\mu(C)$, where $C \subset \mathbb{S}^1$, is a particular subset with positive measure. I would like to know if in any way $\mu(C)$ is related to the following quantity: $$\sup_{x \in \mathbb{S}^1}\mathbb{E}_x[\tau^C(\omega,x)] $$ where $$\tau^C(\omega,x):= \min\{ n \ge 1 \colon X_n(\omega,x) \in C\} $$ is defined for $x \in C$. Here $ X_n(\omega,x)$ is the position of the markov chain at time $n$ with initial condition $x$ and randomness $\omega$. In particular, I would like to know if some inequality of the sort $$\mu(C) \ge\sup_{x \in \mathbb{S}^1} \frac{1}{\mathbb{E}_x[\tau^C(\omega,x)]} $$ holds or not.

The reason for this question is because it is well known that for a Markov Chain in a discrete state space it is possible to recover the invariant measure via the positive recurrent states. I was wandering if something like this was possible too in this case. Thanks in advance.