Question

I was looking for some good references for properties/theorems/characterizations of 'good/important' probability measures on the unit circle $S^1$ ( and/or on spheres $S^n$ ).In particular, I want some probability measures on $S^1$ which is obtained through a kernel, like Poisson kernel for the harmonic measures etc. Are there any theorems that the probabilty measures are generated by some kernel if they are 'good', i.e. they have no atoms, they may be absolutely continuous with respect to Lebesgue measure.

Please suggest me some references in this topic.

Also, are there good results regarding homeomorphisms of $S^1, S^n$ which are more explicit in nature/expression ?

Answer 1

Perhaps this is not exactly what you are expecting, but it is anyway interesting.

In topological dynamics, when you have a compact group $\Gamma$ acting on the circle with Haar measure $dg$, you define a $\Gamma$-invariant probability measure on $S^1$ by the formula $$\mu(A):=\int_\Gamma Leb(gA)dg$$ where $A$ is any Borel subset of $S^1$ and $Leb$ denotes the Lebesgue measure on the circle.

This is an easy exercise, but if you want to find more, one possible reference is Navas' survey on Groups of circle diffeomorphisms

Answer 2

There are some results and further references in Section 7.15 of K. Ito and H.P. McKean, Diffusion Processes and their Sample Paths, Springer, 1965.