Rigidity for Isometric Actions

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 {(X,d)} is a compact metric space and {T:X\rightarrow X} is an isometry. Suppose further that {T} has a dense orbit. Then there exists a compact abelian group {G} and a homeomorphism {\phi:X\rightarrow G} such that {\phi\circ T\circ \phi^{-1}} is a translation by some fixed element {g_0\in G}.

Proof: Suppose {x\in X} has a dense orbit denoted {Y=\{T^i x\}_{i\in {\mathbb Z}}}. Define a map {m:Y\times Y\rightarrow Y} by

\displaystyle  m(T^nx,T^mx)=T^{n+m}x

This defines a group structure on {Y}, and it suffices to check that it extends continuously to all of {X} (dealing with map defining the inverse is similar). Introduce the notation {x_n:=T^nx}. To check the continuity of {m}, it suffices to verify that if {d(x_{n_1},x_{n_2})<\epsilon} and {d(x_{m_1},x_{m_2})<\epsilon}, then {d(x_{n_1+m_1},x_{n_2+m_2})<\epsilon}. However, consider the point {x_{n_1+m_2}}. Because {T} is an isometry, we have

\displaystyle d(x_{n_1+m_1},x_{n_1+m_2})=d(x_{m_1},x_{m_2})<\epsilon

Similarly

\displaystyle  d(x_{n_1+m_2},x_{n_2+m_2})=d(x_{n_1},x_{n_2})<\epsilon

Applying the triangle inequality, we get the desired estimate. \Box

The following result directly follows from the earlier theorem.

Theorem 2 Suppose {M} is a Riemannian manifold and {T:M\rightarrow M} is an isometry. Then for any {x\in M}, the closure of the orbit {\{T^{i}x\}_{i\in{\mathbb Z}}} 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 {X}, with basepoint {x}. Let

\displaystyle  Stab_X=\{g\in Isom(M)|gX=X\}

Then {X=Stab_X/Stab_X\cap Stab_{x}}. This is a manifold, since the stabilizers are closed Lie subgroups of the isometry group. Since {X} is a compact abelian group, it must be a finite union of tori. \Box

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 {G} is a compact group, {\nu} is its Haar probability measure and let {g\in G} be a fixed element. Denote by {R_g} the right translation by {g} on the group. The following are equivalent:

  • (i) {R_g} is ergodic for {\nu}
  • (ii) {G} is abelian and some orbit is dense
  • (iii) {G} is abelian and every orbit is dense
  • (iv) {\nu} is the unique Borel probability measure invariant under {R_g}
  • Proof: {(i)\Rightarrow (ii)} A dense orbit exists by ergodicity, and the continuous map {[,]:G\times G\rightarrow G} defined by {[x,y]=xyx^{-1}y^{-1}} maps to the identity on a dense subset. Thus it maps everything to the identity, so {G} is abelian. {(ii)\Rightarrow (iii)} A dense orbit will visit any neighborhood of any point {x}. Right-translating by {g} to the neighborhood of any other point, any other {R_g} orbit is dense. {(iii)\Rightarrow (iv)} If a measure {\mu} is invariant under {R_g}, it is invariant under {R_{g^n}}, but {{g^n}} will form a dense set in {G}, so {\mu} will be invariant under the entire group. This property characterizes Haar measure. {(iv)\Rightarrow (i)} This is the definition of
    ergodicity. \Box

    Another result which is worth stating but whose proof will not be given is the following.

    Theorem 4 Suppose {M} is a compact Riemannian manifold and {T:M\rightarrow M} is an isometry. Suppose further that {T} has a dense orbit. Then the action of {T} is minimal and uniquely ergodic (with respect to the Riemannian volume).

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