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?

As an example (on a non-compact manifold), there is a theorem by Kazhdan (Uniform distribution on a plane) that concerns the action on the Euclidean plane of the semigroup generated by a rotation and an arbitrary motion.

Nevertheless, 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.

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...).

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 $n$ 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?

As an example (on a non-compact manifold), there is a theorem by Kazhdan (Uniform distribution on a plane) that concerns the action on the Euclidean plane of the semigroup generated by a rotation and an arbitrary motion.

Nevertheless, 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.

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?

As an example (on a non-compact manifold), there is a theorem by Kazhdan (Uniform distribution on a plane) that concerns the action on the Euclidean plane of the semigroup generated by a rotation and an arbitrary motion.

Nevertheless, 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.