Let $p,q$ be two positive integers such that $\frac{\log p}{\log q}\notin\mathbb{Q}$. Furstenberg's $\times p,\times q$ conjecture says that the only ergodic nonatomic $\times p,\times q$-invariant Borel probability on the unit circle $\mathbb{T}$ is the Lebesgue measure.

Just from my personal point of view, the reason that Furstenberg gives his conjecture is the following:

Theorem [Furstenberg, 1967]

A closed $\times p,\times q$-invariant subset of $\mathbb{T}$ is either finite or $\mathbb{T}$.

Motivated by Furstenberg's conjecture, one may wonder the following:

Suppose that a discrete amenable group $\Gamma$ acts on a compact metrizable space $X$ with infinitely many points by homeomorphisms. For every $x\in X$, the orbit $\{\gamma\cdot x\}_{\gamma\in\Gamma}$ is either finite or dense in $X$.

Question: Is there any dynamical system satisfying the above property and having at least two ergodic non-atomic invariant measures?