I'm an REU student who has just recently been thrown into a dynamical system problem without basically any background in the subject. My project advisor has told me that I should represent regions of my dynamical system by letters and look at the sequence of letters formed by the trajectory of a point under the iteration of my map.

He claims that it's a common result that if two points share the same sequence, then this sequence of letters is periodic. I've asked around among some of the other students, and they said that this is sometimes called symbolic dynamics, but none of them remembers this sort of result. I've also searched the internet, but it's possible that my google-fu is weak, since I didn't find any answers that way.

To go one step further, there are obvious cases where it is false- take $S^1\times I$, and encode the regions as $A$ corresponds to $[0,\pi)\times I$ and $B$ corresponds to $[\pi,2\pi)\times I$ with map $f(x,y)=(x+1\mod{2\pi},y)$. Obviously any two points $(x,y)$ and $(x,z)$ with $y\neq z$ will have the same sequence, but since 1 is an irrational multiple of $2\pi$, the trajectory will never be periodic.

I'm interested in the general theory and common techniques applied to the question:

Represent a dynamical system by associating symbols with regions of the space. When is it true that if two distinct points's trajectories have the same sequence of symbols, then the sequence of symbols is periodic?

Any answers, examples, or specific references would be greatly appreciated.

  • $\begingroup$ There are five references listed at the Wikipedia article for symbolic dynamics. Have you tried them? $\endgroup$ – Qiaochu Yuan Jul 22 '11 at 2:07
  • $\begingroup$ What sort of system are you studying? Symbolic dynamics works well for studying systems with some sort of expansivity property, which is why your example has the undesirable behaviour you describe; it's not expanding at all in the second coordinate. $\endgroup$ – Vaughn Climenhaga Jul 22 '11 at 2:32
  • $\begingroup$ @Qiaochu Yuan: I've read the Morse article and I have started reading the Lind and Marcus book, but I'm having trouble identifying useful ideas. @Vaughn Climenhaga: I'm studying outer billiards in the hyperbolic plane (decent reference: math.psu.edu/tabachni/prints/dogtab.pdf). Generally, we use one of the Poincare models of the hyperbolic plane, and we code regions by which vertex they use to reflect. Nothing so bad as our example happens- all orbits except 2 appear bounded and to vary their distance from the center quite a bit. $\endgroup$ – KReiser Jul 22 '11 at 3:08

Intuitively this should happen for a large class of dynamical systems, but I don't know the right necessary and sufficient conditions.

A class of examples satisfying this is given by polyhedral billiards, where you assign a symbol to each face and correspond orbits to sequences in the obvious manner. It is a result of G Galperin, T Krüger and S Troubetzkoy in their paper "Local instability of orbits in polygonal and polyhedral billiards" (1995), that if a trajectory has a periodic sequence then it must be a periodic trajectory and that if there are two different trajectories with the same symbolic sequences then the symbolic sequences are periodic.


I think symbolic dynamics is the study of what you get after you've introduced your partition and coded points by their itinerary. Your question is essentially "when is the symbolic dynamics a faithful representation of the original dynamical system?".

As Vaughn says, the thing you're looking for is expansiveness - effectively a guarantee that if you start with two different points then sooner or later they end up in a different element of the partition. It sounds as though the paper mentioned by Gjergji is very close to what you're looking for.

Other explicit examples where the symbolic dynamics represents things faithfully: Axiom A maps (Smale); Geodesic flows (Bowen and Ratner); Expanding maps. More generally there is a beautiful paper by Boyle and Lind "expansive subdynamics" on expansiveness for higher-dimensional actions (when you don't just have a single transformation acting but a bunch of commuting transformations).


I am not an expert in this topic. I would guess that "minimality" is required to make anything work. For example, take any system with any labelling (also called a partition, or Markov partition if it satisfies various properties). Take any point $x_0$. Let $x_i$ be the $i$-th iterate of $x_0$. Attach a blob $B_i$ to $x_i$, where all of the $B_i$ are identical. Make a new system where $B_i$ gets sent to $B_{i+1}$ and the points of $B_i$ get the label of $x_i$.

Here is a reference for minimality.



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