Uniform distribution of points on Riemannian manifolds

Question

Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem:

Theorem: Let A and B be two rotations of the sphere $S_2$ and let $x$ be a point of the sphere. If the sequence of points $$x, Ax, Bx, A^2x, ABx, BAx, B^2x, \ldots $$ is dense on the sphere, then it is uniformly distributed.

Here, uniformly distributed means that, if we consider the $2^n$ points produced with exactly $n$ iterations of the rotations, namely $$ A^nx , A^{n-1}Bx , A^{n-2}BAx , \ldots , B^nx \ , $$ and we count the proportion of them that lie inside a region $\Delta$ (bounded by a piecewise smooth curve), then these ratios converge to the measure of $\Delta$; that is to say: $$ \lim_{n\to \infty} \frac{\mbox{Number of points among } \{A^nx , \ldots , B^nx \} \mbox{ inside } \Delta}{2^n} = \frac{\mu(\Delta)}{\mu(S_2)}.$$

A quick search has shown to me that there have been many developments of the ideas behind this result, in many different directions.

But I was wondering what is the state of the art concerning the uniform distribution of points under the action of semigroups of isometries in a more general situation.

To be precise,

Question: Is there a similar result for the action of a (finitely generated) semigroup of isometries of a compact Riemannian manifold?

Most of the references I see are too specialized for me and, although they seem very general, I am not able to understand whether they cover the situation I am interested in or not.

Edit 1

In the following reference:

Guivarc'h, Y.: Equirépartition dans les espaces homogènes, in Théorie Ergodique, LNM 532, Springer, Berlin (1976) pp. 131--142.

the author seems to comment in the introduction that an analogous result to that of Arnol'd-Krylov holds for the sequence of probabilities $p_n = \frac{1}{n} \sum_{k=0}^{n-1} p^k$ on a compact Lie group.

Here, $p$ is a sum of Dirac deltas, its power $p^k$ is made via the convolution product and the equidistribution statement is written in terms of convergence of probabilities.

As the group of isometries of a compact Riemannian manifold is a compact Lie group, I assume that the answer to my question is yes, but I am not sure yet (there are many assumptions on Guivarc'h's note and very few details...).

Answer 1

Let $\mu=(\delta_A+\delta_B)/2$. Then the claim of Arnold - Krylov is the weak convergence of the convolutions $\mu^{*n}*\delta_x$ to the rotation invariant probability measure on the sphere (where $\mu^{*n}$ is the $n$-th convolution power of the probability measure $\mu$ on the group of rotations). A general answer to this question had been given by Stromberg Stromberg (1960) several years before Arnold - Krylov (they were not aware of this work) and largely goes back to Kawada - Ito (1940). According to Stromberg's Main Theorem, the sequence of convolution powers of a probability measure $\mu$ on a compact group $K$ weakly converges to the Haar measure $m_K$ if and only if the support of $\mu$ is not contained in a coset of a proper closed normal subgroup.

EDIT The condition on the support of $\mu$ is obviously necessary as otherwise (if $\mu(gH)=1$ for a proper closed normal subgroup $H\subset K$) the image of $\mu$ under the quotient map $G\to G/H$ is concentrated on a single element, and therefore the image of $\mu^{*n}$ is the $n$-th power of this element. As for the original question, the point is that the group of isometries of a compact Riemannian manifold is also compact. There does exist "Haar measure" (or, rather, measures) on general compact manifolds - these are the unique invariant measures on the orbits of the group of isometries. Stromberg's theorem describes the limits of convolution powers on the group of isometries, and therefore, on its homogeneous spaces as well.

If you are interested in the convergence of the Cesaro averages rather than convolution powers themselves, condition $\mu(gH)<1$ is no longer necessary, and $(\mu +\dots+ \mu^{*n})/n$ converges to the Haar measure on the subgroup generated by the support for any measure $\mu$ on a compact group $K$. All this essentially goes back to Ito -- Kawada, and a good account can be found in old Grenander's book.