A generalized Furstenberg's $\times p,\times q$-conjecture

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?

• Beware that on the unit circle, it makes little sense to call this $\times p,\times q$, but rather $\hat{}p,\hat{}q$. $\times p,\times q$ makes sense on the additive model of the circle $\mathbf{R}/t\mathbf{Z}$. – YCor Nov 23 '18 at 6:37
• Why do you call this "generalized" in the title? please also specify what you call "dynamical system" in your question (what is acting? an element, an invertible element, a semigroup, a group?) Could you start with your question, and then possibly add context. – YCor Nov 23 '18 at 6:39
• It’s well known that there are minimal sub shifts that are not uniquely ergodic. These form a counterexample to your conjecture – Anthony Quas Nov 23 '18 at 9:13
• Understanding Anthony's answer: first, this was not a generalization of the conjecture because what's acting in Furstenberg's dynamical system is a semigroup (free abelian on 2 generators) and not a group. And of course Furstenberg was aware of these non-uniquely ergodic examples in symbolic dynamics, and certainly made his conjecture on more grounds. – YCor Nov 23 '18 at 10:08
• Most people think that Furstenberg's conjecture is true. But I am not clear where the belief comes from. That's the reason that I ask the question. And it's well-known that the $\times p,\times q$ action on the unit circle could be lifted to a $\mathbb{Z}^2$ action on the pq-solenoid. The sets of ergodic invariant measures on these two dynamical systmes are homeomorphic. – Huichi Huang Nov 24 '18 at 3:30