## Distal systems and Expansive systems

A topological dynamical system is a pair ${(X,T)}$ where ${X}$ is a compact metric space and ${T:X\rightarrow X}$ is a continuous map. This gives rise to an action of the semigroup ${({\mathbb N},+)}$ on ${X}$, where ${n\in{\mathbb N}}$ acts via the iterated map ${T^n=T\circ T^{n-1}}$ (and as usual ${T^0}$ denotes the identity map). If ${T}$ is invertible (i.e. a homeomorphism), then in fact it induces a ${({\mathbb Z},+)}$-action. More generally, one may consider an action ${T=(T_g)_{g\in G}}$ of any semigroup ${G}$ on ${X}$ by continuous functions. This means that for each ${g\in G}$ we have a continuous map ${T_g:X\rightarrow X}$ which satisfy the semigroup law ${T_g\circ T_h=T_{gh}}$. In this case we say that ${(X,T)}$ is a ${G}$-system. Throughout this post I will denote by ${d_X}$ (or ${d}$ if the underlying space is clear) a compatible metric on ${X}$.

Two important classes of topological dynamical systems are the class of distal systems and the class of expansive systems.

Definition 1

• A ${G}$-system ${(X,T)}$ is distal if

$\displaystyle \forall x,y\in X, \ x\neq y,\ \inf\big\{d(T_g x,T_g y):g\in G\big\}>0.$

• A ${G}$-system ${(X,T)}$ is expansive if there exists ${\delta>0}$ such that

$\displaystyle \forall x,y\in X, \ x\neq y,\ \sup\big\{d(T_g x,T_g y):g\in G\big\}>\delta.$

While not entirely trivial, both properties are preserved by topological conjugacy (i.e., isomorphism in the category of ${G}$-systems), and in particular don’t depend on the choice of the (compatible) metric ${d}$.

At a first glance at the definition it looks like the two conditions are quite similar. They both fall into the loose statement that “if two points are distinct, then they are far apart in the future” (at least if the acting semigroup is ${{\mathbb N}}$). However, it turns out that the two properties are very much different, and in some sense actually incompatible.

In this post I will mention some examples and properties of distal and expansive systems which illustrate this difference between the two classes, and present a proof of the incompatibility result:

Theorem 2 An ${{\mathbb Z}}$-system which is both distal and expansive must be finite. The same is true for ${{\mathbb N}}$-systems.

It is clear that any finite system (i.e., where ${X}$ is a finite set) is both distal and expansive.

— 1. Distal systems —

An example of a distal ${{\mathbb N}}$-system is a rotation on the circle group ${{\mathbb T}={\mathbb R}/{\mathbb Z}}$ induced by the map ${T:x\mapsto x+\alpha}$ for a fixed ${\alpha\in{\mathbb T}}$. Indeed, ${T}$ is an isometry (i.e. ${d(Tx,Ty)=d(x,y)}$ for all ${x,y\in X}$). In general, if ${(X,T)}$ is a ${G}$-system where for each ${g\in G}$ the map ${T_g}$ is an isometry on ${X}$, then for any points ${x,y\in X}$ one has ${\inf\{d(T_gx,T_gy):g\in G\}=d(x,y)}$, and hence ${(X,T)}$ is distal.

However, it is difficult for an isometric ${G}$-system to be expansive. Indeed, if ${T}$ is an action by isometries and ${(X,T)}$ is expansive, then for any ${x,y}$ with ${x\neq y}$ we have ${d(x,y)>\delta}$, for some constant ${\delta>0}$. Since ${X}$ is compact, we conclude that ${X}$ must be finite. This already sheds some light on the difference between distality and expansiveness.

Isometric systems are closely connected with group rotations. A topological dynamical system ${(X,T)}$ is called transitive if there exists a dense orbit (this is usually a benign assumption to make on a system since one it often interested on the orbit of a single point).

Proposition 3 Let ${(X,T)}$ be a transitive ${{\mathbb N}}$-system where ${T}$ is an isometry. Then ${X}$ can be given a (compatible) topological (abelian) group structure so that ${T:x\mapsto ax}$ for some ${a\in X}$.

Proof: Let ${x\in X}$ be such that the orbit ${\{T^nx:n\in{\mathbb N}\}}$ is dense in ${X}$. For ${y,z\in X}$ find sequences ${(n_i)}$ and ${(m_i)}$ such that ${T^{n_i}x\rightarrow y}$ and ${T^{m_i}x\rightarrow z}$. Define a group operation on ${X}$ via ${y*z:=\lim T^{n_i+m_i}x}$. Observe that the limit defining ${y*z}$ exists; indeed the sequence ${x_i=T^{n_i+m_i}x}$ is Cauchy because for ${i>j}$ we have ${d(x_i,x_j)=d(T^{n_i-n_j+m_i-m_j}x,x)\leq d(T^{n_i-n_j}x,x)+d(T^{m_i-m_j}x,x)}$ and this tends to ${0}$ as ${j\rightarrow\infty}$. Next observe that ${y*z}$ does not depend on the choice of the sequence ${(n_i)}$. Indeed, if ${(\tilde n_i)}$ is another sequence with ${T^{\tilde n_i}x\rightarrow y}$, then ${d(T^{|\tilde n_i-n_i|}x,x)\rightarrow0}$ as ${i\rightarrow\infty}$. These observations show that the operation ${y*z}$ is well defined. It is routine to check that ${(X,*)}$ is indeed a topological group (with the topology induced by the metric ${d}$) with identity ${x}$. Let ${a\in X}$ be the element ${a=Tx}$, it is now easy to see that ${Ty=a*y}$ for every ${y\in X}$. $\Box$

Note that, as a corollary of Proposition 3, we obtain the fact that every isometry of a compact metric space is in fact invertible, i.e., a homeomorphism. In other words, the ${{\mathbb N}}$-system ${(X,T)}$ can be extended to a ${{\mathbb Z}}$-system. More generally, if ${G}$ is a group and ${S\subset G}$ is a semigroup which generates ${G}$, then any ${S}$-action by isometries extends to a ${G}$-action. This turns out to be true for distal systems as well (this follows from Theorem 3.1 in this paper of Furstenberg.)

It makes sense to think of isometric systems as very “rigid”, as opposed to “chaotic”. For instance if two points are “nearby” at some time, then they remain “nearby” at all times.

A natural construction in topological dynamics is that of a group extension. We formulate this construction only in the case of one transformation (i.e., where the acting semigroup is ${{\mathbb N}}$) to simplify the notation and details.

Definition 4 Let ${(X,T)}$ be a ${{\mathbb N}}$-system, let ${K}$ be a compact abelian group and let ${Y=X\times K}$. Given a continuous map ${\phi:X\rightarrow K}$, let ${S:Y\rightarrow Y}$ be defined by ${S(x,z)=(Tx,z+\phi(x))}$. We say that ${(Y,S)}$ is a group extension of ${(X,T)}$.

An example of a group extension of the rotation ${(X,T)}$ where ${X={\mathbb T}}$ and ${T:x\mapsto x+\alpha}$ for a fixed ${\alpha\in{\mathbb T}}$ is the algebraic skew-product ${(Y,S)}$ where ${Y={\mathbb T}^2}$ and ${S:(x,z)\mapsto(x+\alpha,y+x)}$. Group extensions, in general, do not add too much “complexity” to a system, since each fiber of the extension behaves like a group rotation, which as seen above is an isometry.

It turns out that a group extension of a distal system is also distal.

Proposition 5 Let ${(X,T)}$ be a distal ${{\mathbb N}}$-system (or, equivalently, a distal ${{\mathbb Z}}$-system) and let ${(Y,S)}$ be a group extension of ${(X,T)}$. Then ${(Y,S)}$ is distal.

Proof: Write ${Y=X\times K}$ for a compact abelian group ${K}$ and let ${\phi:X\rightarrow K}$ be such that ${S(x,z)=(Tx,z+\phi(x))}$. Also let ${\pi_X:Y\rightarrow X}$ and ${\pi_K:Y\rightarrow K}$ be the natural projections. Given ${y_1,y_2\in Y}$ distinct, write ${y_i=(x_i,z_i)}$ for ${i=1,2}$. After choosing metrics satisfying ${d_Y(y_1,y_2)=d_X(x_1,x_2)+d_K(z_1,z_2)}$ for every ${y_i=(x_i,z_i)\in Y}$, we have ${d_Y(S^ny_1,S^ny_2)\geq d_X(T^nx_1,T^nx_2)}$ for every ${n}$.

If ${x_1\neq x_2}$ then, since ${(X,T)}$ is distal, it follows that ${\inf_n d_Y(S^ny_1,S^ny_2)>0}$.

If ${x_1=x_2}$, then ${z_1\neq z_2}$. In this case, ${\pi_X(S^ny_1)=T^nx_1=\pi_X(S^ny_2)}$ for all ${n}$, so ${d_Y(S^ny_1,S^ny_2)=d_K(\pi_K(S^ny_1),\pi_K(S^ny_2))}$. Since ${\pi_K(Sy)=y+\phi(\pi_X(y))}$ it follows that ${d_K(\pi_K(Sy),\pi_K(Sy'))=d_K(\pi_K(y),\pi_K(y'))}$ as long as ${\pi_X(y)=\pi_X(y')}$. Iterating this observation it follows that ${d_K(\pi_K(S^ny_1),\pi_K(S^ny_2))=d_K(z_1,z_2)>0}$ for all ${n\in{\mathbb N}}$. $\Box$

A remarkable result of Furstenberg states that all transitive distal systems are essentially obtained in this way. In fact, Furstenberg’s theorem applies to ${G}$-systems for arbitrary countable group ${G}$. The theorem involves the notion of isometric extension, which is a somewhat more complicated analogue of group extensions.

Theorem 6 Fix a countable group ${G}$. The class of all distal ${G}$-systems is the smallest class containing the trivial one point system and closed under isometric extensions and inverse limits (in the category of ${G}$-systems).

Here are some other facts about distal systems:

Theorem 7 Let ${(X,T)}$ be a distal ${G}$-system. Then

• ${(X,T)}$ admits an invariant measure. (this is Theorem 12.3 in Furstenberg’s paper)
• If ${G={\mathbb Z}}$, then ${(X,T)}$ has ${0}$ (topological) entropy. (this is, for instance, Corollary 18.21 in Glasner’s book)
• ${(X,T)}$ is semi-simple, i.e. it is the disjoint union of minimal systems (this is Theorem 3.2 in Furstenberg’s paper)
• For any other distal ${G}$-system ${(Y,S)}$, the product system ${(X\times Y,T\times S)}$ is distal. (this is easy to check directly)
• In fact ${(X,T)}$ is distal if and only if the self product ${(X\times X,T\times T)}$ is semisimple. (this is also not too difficult to check directly)
• Any factor of ${(X,T)}$ is distal. (this follows from the previous point)
• If ${G}$ is abelian and ${(X,T)}$ is transitive and non-trivial, then it has a nonconstant eigenfunction. (this is proved in Section 11 in Furstenberg’s paper)

— 2. Expansive systems —

A good example of an expansive system to have in mind is the ${{\mathbb N}}$-system induced by the doubling map ${T:x\mapsto2x}$ on the circle group ${{\mathbb T}}$. In fact, if two points ${x,y}$ are very close together, then ${2x,2y}$ are twice as far apart as ${x}$ and ${y}$. More precisely, if ${d(x,y)<1/4}$ then ${d(2x,2y)=2d(x,y)}$. This shows that for any distinct ${x,y\in{\mathbb T}}$ there is some ${n\in{\mathbb N}}$ with ${d(T^nx,T^ny)\geq1/4}$. (One can in fact replace ${1/4}$ in the last sentence with ${1/3}$, but that takes some more work to justify. Taking ${x=1/3}$ and ${y=2/3}$ one sees that ${1/3}$ is the optimal constant.) Since the points ${0}$ and ${1/2}$ are both mapped to ${0}$, it follows that this system is not distal.

An interesting class of examples of expansive systems is provided by subshifts, which we presently define. Let ${G}$ be a countable semigroup. Given a finite set ${A}$, let ${X=A^G}$ and let ${T_g:X\rightarrow X}$ be the shift ${T_g:(x_h)_{h\in G}\mapsto(x_{hg})_{h\in G}}$. Giving ${A}$ the discrete topology and ${X}$ the product topology, the pair ${(X,T)}$ forms a ${G}$-system, called the full shift. A compatible metric is given by enumerating ${G=\{g_1,g_2,\dots\}}$ and letting ${d(x,y)=1/\min\{i:x_{g_i}\neq y_{g_i}\}}$.

Assume that ${G}$ is a group. If ${x,y\in X}$ are distinct, and ${i\in{\mathbb N}}$ is such that ${x_{g_i}\neq y_{g_i}}$, then ${d(T_{g_1^{-1}g_i}x,T_{g_1^{-1}g_i}y)=1}$. This shows that a full shift ${(X,T)}$ is expansive (this proof doesn’t seem to work for general semigroups, but the result holds if ${G}$ is finitely generated, or if it has an identity. However, it is not true in general, see Example 1 below). One should think of the full shift as a very chaotic system. Indeed, we can fabricate any kind of behaviour, for instance, two points that start very close and end up far apart. Or two distinct points that get closer and closer to each other, colliding after infinite time.

Example 1 Let ${G={\mathbb N}\setminus\{1\}}$ be the semigroup of integers larger than ${1}$ with the usual multiplication. Then a full shift ${(X,T)}$ over ${G}$ is not expansive.

Proof: I claim that for any compatible metric ${d}$ and any ${\delta>0}$ there is a finite set ${F\subset G}$ such that for any ${x,y\in X}$, if ${x(n)=y(n)}$ for all ${n\in F}$, then ${d(x,y)<\delta}$. Assuming the claim for now, let ${p}$ be a prime which does not divide any element of ${F}$ and let ${x,y\in X}$ be such that ${x(n)=y(n)}$ for all ${n\neq p}$, and ${x(p)\neq y(p)}$. Then ${T_mx(n)=T_my(n)}$ for all ${n\in F}$ and ${m\in G}$ (because ${T_mx(n)=x(nm)}$ and ${nm\neq p}$ whenever ${n\in F}$ and ${m\in G}$). We conclude that ${d(T_m x,T_m y)0}$. For each ${z\in X}$ let ${C_z}$ be a cylinder set containing ${z}$ and contained in the open ball around ${z}$ with radius ${\delta/2}$. By compactness, there is a finite set ${Z\subset X}$ such that ${\{C_z:z\in Z\}}$ is a cover of ${X}$. For each ${z\in Z}$ let ${F_z\subset G}$ be such that ${C_z=\{x\in X:x(n)=z(n)\text{ for all }n\in F_z\}}$. Let ${F=\bigcup_{z\in Z} F_z}$. Then given ${x,y\in X}$ with ${x(n)=y(n)}$ for all ${n\in F}$, let ${z\in Z}$ be such that ${x\in C_z}$. Since ${x(n)=y(n)}$ for all ${n\in F_z}$ it follows that ${y\in C_z}$. Therefore ${d(x,y). $\Box$

A subshift is the system ${(Y,T)}$ where ${Y}$ is a closed subset of ${X}$ which is invariant under the shift action ${T}$ in the sense that ${T_g(Y)\subset Y}$ for all ${g\in G}$. The same argument presented above shows that

Proposition 8 If ${G}$ is a group, then every subshift is expansive.

It follows from Furstenberg classification of transitive distal systems mentioned in the previous section that a distal system has always a non-trivial factor which is an isometry, in a sense the most well behaved kind of dynamical system. It turns out that general expansive system are also connected to the most chaotic kind of systems (i.e. subshifts) via a factor map; however the factor map goes in the opposite direction.

Proposition 9 (Reddy) Let ${(X,T)}$ be an expansive ${G}$-system, where ${G}$ is a group. Then it is a factor of a subshift ${(Y,S)}$.

Proof: Let ${\delta>0}$ be such that for every distinct ${x,y\in X}$ there is ${g\in G}$ with ${d(T_gx,T_gy)>\delta}$. Let ${A}$ be a finite open cover of ${X}$ by balls of diameter ${\delta}$. Then for every distinct ${x,x'\in X}$ there exists ${g\in G}$ such that ${T_gx}$ and ${T_gx'}$ fall in different elements of ${A}$.

Let ${Y\subset A^G}$ be the set of points ${y=(y_g)_{g\in G}}$ for which there exists ${x=x(y)\in X}$ with ${T_gx\in\overline{y_g}}$ for all ${g\in G}$ (recall that ${y_g}$ is an element of ${A}$ and hence a subset of ${X}$). Observe that such ${x}$, if it exists, must be unique; call it ${\Psi(y)}$. Since the complement of ${Y}$ in ${A^G}$ is a union of cylinders (indeed, a point ${z\in A^G}$ is not in ${Y}$ precisely if ${\bigcap_{g\in G}T_g^{-1}\overline{z_g}=\emptyset}$; but for this to happen, some finite intersection must already be empty), ${Y}$ is closed and hence compact. Clearly ${\Psi:Y\rightarrow X}$ is surjective, we claim that it is continuous. To prove the claim let ${y^{(k)}}$ be a sequence in ${Y}$ such that ${y^{(k)}\rightarrow y}$ in ${Y}$. Passing to a subsequence if necessary we can assume that the sequence ${x_k:=\Psi(y^{(k)})}$ converges to some point ${x\in X}$ and we need to show that ${\Psi(y)=x}$. This follows if we show that for every ${g\in G}$ we have ${T_gx\in\overline{y_g}}$, which is equivalent to ${x\in T_g^{-1}\overline{y_g}}$. But for any fixed ${g\in G}$, if ${k}$ is large enough (depending on ${g}$) we have ${y_g=y_g^{(k)}}$, so ${x_k\in T_g^{-1}\overline{y_g}}$ for all large ${k}$, which implies that ${T_gx\in\overline{y_g}}$, proving the claim.

Finally, letting ${S}$ be shift action on ${Y}$ it follows from the definition of ${\psi}$ that ${\Psi\circ S_g=T_g\circ\Psi}$ for all ${g\in G}$, finishing the proof. $\Box$

In the previous section we saw that a distal ${{\mathbb N}}$-system can always be extended to a (distal) ${{\mathbb Z}}$-system. The situation with expansive systems is essentially the opposite.

Proposition 10 Let ${(X,T)}$ be an expansive ${{\mathbb N}}$-system which can be extended to a ${{\mathbb Z}}$-system (i.e., so that ${T}$ is a homeomorphism). Then ${X}$ is a finite set.

Observe that Proposition 10 already yields a proof of the incompatibility result Theorem 2 for ${{\mathbb N}}$-systems. The next theorem covers the case of ${{\mathbb Z}}$-systems.

Theorem 11 Let ${(X,T)}$ be a ${{\mathbb Z}}$-system which is distal and expansive. Then ${X}$ is a finite set.

Proof: This proof is essentially due to Utz. Suppose that ${X}$ is infinite and the system ${(X,T)}$ is expansive. We will show that ${(X,T)}$ can not be distal.

Since ${X}$ is infinite, using compactness we can find a sequence ${(x_n)_{n\in{\mathbb N}}}$ in ${X}$ which converges to a point ${x\in X}$ and such that ${x_n\neq x}$ for all ${n\in{\mathbb N}}$. We can then use the expansive assumption to find ${\delta>0}$ and integers ${m_n}$ such that ${d(T^{m_n}x_n,T^{m_n}x)>\delta}$ for all ${n\in{\mathbb N}}$. We can (and will) assume without loss of generality that for every ${k\in{\mathbb Z}}$ with ${|k|<|m_n|}$ we have ${d(T^kx_n,T^kx)\leq\delta}$. Since ${x_n\rightarrow x}$ it follows that ${|m_n|\rightarrow\infty}$ as ${n\rightarrow\infty}$ (because for every fixed ${m}$ the map ${T^m}$ is continuous). Replacing ${T}$ with ${T^{-1}}$ if necessary and passing to a subsequence, we can assume that in fact ${m_n\rightarrow-\infty}$ as ${n\rightarrow\infty}$.

Using compactness, and after passing to a subsequence again if needed we may assume that the limits

$\displaystyle y=\lim_{n\rightarrow\infty}T^{m_n} x_n\text{ and }z=\lim_{n\rightarrow\infty}T^{m_n}x$

both exist. Clearly ${d(y,z)\geq\delta>0}$ so ${y\neq z}$. However for every ${k\in{\mathbb N}}$,

$\displaystyle d(T^ky,T^kz)=\lim_{n\rightarrow\infty}d(T^{m_n+k}x_n,T^{m_n+k}x)\leq\delta$

because $|m_n+k|<|m_n|$ whenever $-m_n\geq k$ (which occurs for all but finitely many $n$). Using again compactness, find a sequence ${(k_i)}$ of natural numbers tending to ${\infty}$ such that the limits

$\displaystyle \tilde y:=\lim_{i\rightarrow\infty}T^{k_i}y\text{ and }\tilde z:=\lim_{i\rightarrow\infty}T^{k_i}z$

both exist. We claim that ${\tilde y=\tilde z}$, which shows that the ${(X,T)}$ is not distal. To prove the claim, note that for any ${s\in{\mathbb Z}}$ we have

$\displaystyle d(T^s\tilde y,T^s\tilde z)=\lim_{i\rightarrow\infty}d(T^{k_i+s}y,T^{k_i+s}z)\leq\delta$

since ${k_i+s}$ is positive for all sufficiently large ${i}$. Therefore ${\tilde y=\tilde z}$ follows from directly from the fact that ${(X,T)}$ is expansive. $\Box$

It seems this proof does not work for other groups, or semigroups. I do not know if the result is true in that generality.

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### 2 Responses to Distal systems and Expansive systems

1. Sihan says:

It’s definitely a comprehensive guide to the concepts of expansive and distal systems.
However, I am wondering that, in the last proof of the Theorem 11, how can we deduce the following:

For every natural number $k$, the distance of $T^k(y)$ and $T^k(z)$ is no greater than $\delta$? Since, according to the assumption, this happens if $|m_n+k|<|m_n|$.

Thank you so much.

• Joel Moreira says:

Dear Sihan,

There were some typos (cause by html confusing the symbol “<" for a command) which make the sentence difficult to decipher, I've now fixed the issue.
In any case, the point here is that $m_n\to-\infty$ as $n\to\infty$, so indeed for every fixed $k$, eventually we have $|m_n+k|<|m_n|$.