A topological dynamical system is a pair where is a compact metric space and is a continuous map. This gives rise to an action of the semigroup on , where acts via the iterated map (and as usual denotes the identity map). If is invertible (i.e. a homeomorphism), then in fact it induces a -action. More generally, one may consider an action of any semigroup on by continuous functions. This means that for each we have a continuous map which satisfy the semigroup law . In this case we say that is a -system. Throughout this post I will denote by (or if the underlying space is clear) a compatible metric on .
Two important classes of topological dynamical systems are the class of distal systems and the class of expansive systems.
While not entirely trivial, both properties are preserved by topological conjugacy (i.e., isomorphism in the category of -systems), and in particular don’t depend on the choice of the (compatible) metric .
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 ). 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:
It is clear that any finite system (i.e., where is a finite set) is both distal and expansive.
— 1. Distal systems —
An example of a distal -system is a rotation on the circle group induced by the map for a fixed . Indeed, is an isometry (i.e. for all ). In general, if is a -system where for each the map is an isometry on , then for any points one has , and hence is distal.
However, it is difficult for an isometric -system to be expansive. Indeed, if is an action by isometries and is expansive, then for any with we have , for some constant . Since is compact, we conclude that 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 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).
Proof: Let be such that the orbit is dense in . For find sequences and such that and . Define a group operation on via . Observe that the limit defining exists; indeed the sequence is Cauchy because for we have and this tends to as . Next observe that does not depend on the choice of the sequence . Indeed, if is another sequence with , then as . These observations show that the operation is well defined. It is routine to check that is indeed a topological group (with the topology induced by the metric ) with identity . Let be the element , it is now easy to see that for every .
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 -system can be extended to a -system. More generally, if is a group and is a semigroup which generates , then any -action by isometries extends to a -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 ) to simplify the notation and details.
An example of a group extension of the rotation where and for a fixed is the algebraic skew-product where and . 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.
Proof: Write for a compact abelian group and let be such that . Also let and be the natural projections. Given distinct, write for . After choosing metrics satisfying for every , we have for every .
If then, since is distal, it follows that .
If , then . In this case, for all , so . Since it follows that as long as . Iterating this observation it follows that for all .
A remarkable result of Furstenberg states that all transitive distal systems are essentially obtained in this way. In fact, Furstenberg’s theorem applies to -systems for arbitrary countable group . The theorem involves the notion of isometric extension, which is a somewhat more complicated analogue of group extensions.
Here are some other facts about distal systems:
— 2. Expansive systems —
A good example of an expansive system to have in mind is the -system induced by the doubling map on the circle group . In fact, if two points are very close together, then are twice as far apart as and . More precisely, if then . This shows that for any distinct there is some with . (One can in fact replace in the last sentence with , but that takes some more work to justify. Taking and one sees that is the optimal constant.) Since the points and are both mapped to , 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 be a countable semigroup. Given a finite set , let and let be the shift . Giving the discrete topology and the product topology, the pair forms a -system, called the full shift. A compatible metric is given by enumerating and letting .
Assume that is a group. If are distinct, and is such that , then . This shows that a full shift is expansive (this proof doesn’t seem to work for general semigroups, but the result holds if 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.
Proof: I claim that for any compatible metric and any there is a finite set such that for any , if for all , then . Assuming the claim for now, let be a prime which does not divide any element of and let be such that for all , and . Then for all and (because and whenever and ). We conclude that . For each let be a cylinder set containing and contained in the open ball around with radius . By compactness, there is a finite set such that is a cover of . For each let be such that . Let . Then given with for all , let be such that . Since for all it follows that . Therefore .
A subshift is the system where is a closed subset of which is invariant under the shift action in the sense that for all . The same argument presented above shows that
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.
Proof: Let be such that for every distinct there is with . Let be a finite open cover of by balls of diameter . Then for every distinct there exists such that and fall in different elements of .
Let be the set of points for which there exists with for all (recall that is an element of and hence a subset of ). Observe that such , if it exists, must be unique; call it . Since the complement of in is a union of cylinders (indeed, a point is not in precisely if ; but for this to happen, some finite intersection must already be empty), is closed and hence compact. Clearly is surjective, we claim that it is continuous. To prove the claim let be a sequence in such that in . Passing to a subsequence if necessary we can assume that the sequence converges to some point and we need to show that . This follows if we show that for every we have , which is equivalent to . But for any fixed , if is large enough (depending on ) we have , so for all large , which implies that , proving the claim.
Finally, letting be shift action on it follows from the definition of that for all , finishing the proof.
In the previous section we saw that a distal -system can always be extended to a (distal) -system. The situation with expansive systems is essentially the opposite.
Proof: This proof is essentially due to Utz. Suppose that is infinite and the system is expansive. We will show that can not be distal.
Since is infinite, using compactness we can find a sequence in which converges to a point and such that for all . We can then use the expansive assumption to find and integers such that for all . We can (and will) assume without loss of generality that for every with we have . Since it follows that as (because for every fixed the map is continuous). Replacing with if necessary and passing to a subsequence, we can assume that in fact as .
Using compactness, and after passing to a subsequence again if needed we may assume that the limits
both exist. Clearly so . However for every ,
because whenever (which occurs for all but finitely many ). Using again compactness, find a sequence of natural numbers tending to such that the limits
both exist. We claim that , which shows that the is not distal. To prove the claim, note that for any we have
since is positive for all sufficiently large . Therefore follows from directly from the fact that is expansive.
It seems this proof does not work for other groups, or semigroups. I do not know if the result is true in that generality.