# Systems ds.dynamical: a generalized Furstenberg \$ times p, times q \$ -conjecture

Leave $$p, q$$ be two positive integers such that $$frac { log p} { log q} notin mathbb {Q}$$. From Furstenberg $$times p, times q$$ The conjecture says that the only non-atomic ergodic. $$times p, times q$$-Invariant probability of Borel in the unit circle. $$mathbb {T}$$ It is the measure of Lebesgue.

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

Theorem [Furstenberg, 1967]

A closed $$times p, times q$$-set variant $$mathbb {T}$$ it is finite or $$mathbb {T}$$.

Motivated by the Furstenberg conjecture, one may wonder:

Suppose that a discrete susceptible group $$Gamma$$ Acts on a compact and metrizable space. $$X$$ With infinite points for homeomorphisms. For each $$x in X$$, the orbit $${ gamma cdot x } _ { gamma in Gamma}$$ is finite or dense in $$X$$.

Question: Is there any dynamic system that satisfies the previous property and has at least two invariant non-atomic ergodic measures?