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:



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



Periodic sequences in symbolic dynamics
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.


