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?