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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?

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  • $\begingroup$ 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 $\mathbf{R}/t\mathbf{Z}$. $\endgroup$ – YCor Nov 23 '18 at 6:37
  • $\begingroup$ 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. $\endgroup$ – YCor Nov 23 '18 at 6:39
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    $\begingroup$ It’s well known that there are minimal sub shifts that are not uniquely ergodic. These form a counterexample to your conjecture $\endgroup$ – Anthony Quas Nov 23 '18 at 9:13
  • $\begingroup$ 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. $\endgroup$ – YCor Nov 23 '18 at 10:08
  • $\begingroup$ 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. $\endgroup$ – Huichi Huang Nov 24 '18 at 3:30

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