Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$.
The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. I'm interesting in knowing whether the operator has a unique and asymptotic distribution $f^*$ (think e.g. whether an attractor exists).
Is it enough to prove that the measure $\mu$ is ergodic with respect to $F$ to show that $f^*$ is unique?
If yes: in Markovian stochastic processes (MSP), one way to prove that the Markov process $F$ has a unique asymptotic distribution $f^*$ is to prove that the process is ergodic.
Are these two things connected? I.e. that uniqueness is given by ergodicity in both cases?
If yes: in MSP, the existence of an asymptotic distribution is guaranteed when the process fulfils detailed balance. What would be the analogy of "detailed balanced" in the Perron-Frobenius operator?
If I understand your question correctly, the answer is "no", simply due to the fact that in general a potential may have more than one Gibbs measures (meaning one invariant probability Gibbs measure, since there will always exist many noninvariant if there is one invariant). The truth is that without some hypotheses this is the best you can say. For example, it is very difficult to agree that "natural measure" is the same for everybody.
On the other hand, it is unfortunate that people call "ergodic" to Markov chains that really have stronger properties than ergodicity, which causes that the two things are related, although not the same.