Let $f:(X, d)\to (X, d)$ be a homeomorphism on compact metric space $(X, d)$.
Function $\theta:X\to \mathbb{R}$ is called a Lyapunov function for $f$ if it is a continuous function with $\theta(f(x))\leq \theta (x)$.
Take $N(\theta)=\left\{x: \theta(f(x))= \theta(x)\right\}$.
I would like to know relation between $N(\theta)$ with $\Omega(f)$ ( $\Omega(f)$ is the set of non-wandering points of $f$)
Conley theorem said that for a system $(X, f)$ on compact metric space $(X, d)$,there is a Lyapunov function $\theta$ for $f$ with $\mathcal{CR}(f)=N(\theta)$.
Is it true that for every system $(X, f)$ on compact metric space and every Lyapunov function $\theta$ for $f$, $\mathcal{CR}(f)\subseteq N(\theta)$?