In this post, I will discuss some elementary results about the action of an isometry on a compact metric space. I found them as homework problems assigned by Prof. Amie Wilkinson in a Smooth Dynamics course, and although quite elementary, I found the results rather striking.
In summary, (ergodic) actions by isometries look a lot like translations on compact abelian groups. For example, orbit closures in a Riemannian manifold are always tori. Precise statements are below the fold.
Theorem 1 Suppose is a compact metric space and is an isometry. Suppose further that has a dense orbit. Then there exists a compact abelian group and a homeomorphism such that is a translation by some fixed element .
Proof: Suppose has a dense orbit denoted . Define a map by
This defines a group structure on , and it suffices to check that it extends continuously to all of (dealing with map defining the inverse is similar). Introduce the notation . To check the continuity of , it suffices to verify that if and , then . However, consider the point . Because is an isometry, we have
Applying the triangle inequality, we get the desired estimate.
The following result directly follows from the earlier theorem.
Theorem 2 Suppose is a Riemannian manifold and is an isometry. Then for any , the closure of the orbit is a finite union of tori.
Proof: By the result just proved, we know the orbit closure is a compact abelian group. Denote it by , with basepoint . Let
Then . This is a manifold, since the stabilizers are closed Lie subgroups of the isometry group. Since is a compact abelian group, it must be a finite union of tori.
Remark 1 Note that the above proof uses implicitly the solution to Hilbert’s fifth problem. I am not sure it can be avoided.
At the end, let us formulate several results that describe the ergodic theory of compact groups.
Theorem 3 Suppose is a compact group, is its Haar probability measure and let be a fixed element. Denote by the right translation by on the group. The following are equivalent:
(i) is ergodic for (ii) is abelian and some orbit is dense (iii) is abelian and every orbit is dense (iv) is the unique Borel probability measure invariant under
Proof: A dense orbit exists by ergodicity, and the continuous map defined by maps to the identity on a dense subset. Thus it maps everything to the identity, so is abelian. A dense orbit will visit any neighborhood of any point . Right-translating by to the neighborhood of any other point, any other orbit is dense. If a measure is invariant under , it is invariant under , but will form a dense set in , so will be invariant under the entire group. This property characterizes Haar measure. This is the definition of
Another result which is worth stating but whose proof will not be given is the following.
Theorem 4 Suppose is a compact Riemannian manifold and is an isometry. Suppose further that has a dense orbit. Then the action of is minimal and uniquely ergodic (with respect to the Riemannian volume).