@MoisheKohan Ah yes again Maximal torus the orbit remain there forever

@MoisheKohan what about the following modified question: a connected compact Lie group for which the set of all elements with dense orbit has an intermediate measure

two conjugate 3lements have the sam3 ergodic type but all elements are conjugate to som3 g in T

@MoisheKohan Thank you! I think you are indicating to the Maximal torus and foliati9n of G with this torus. yes?

A possible relation between your question and existence of linear retraction $B(H)\to M$?

I mean that: assume that we have such a retraction then we extend it to a retraction $B(H)\to M$ since every element has a standard expression in terms of unitaries. So what would be happen? Are there relation between these two questions?

Since unitaries generates the whole algebras is it reasonable to ask the similar question for existence of retraction for $M\to B(H)$? Can one imagine a possible relation betwen these two questions?

@YCor I mean in the connected case. However your first coment is a ful answer to my question. But I try to extend the question

@YCor ecause in the Lie group case a subgroup can not have intermediate measure but in you example the subgroup has index 2

@YCor Yes thank you that is perfect. I was inspired by the circle case. The ergodic element has full measure. So what about we add the extra assumption connected lie group? BTW is there an example of empty or zero measure set of ergodic elements?