# 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).