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Symbolic dynamics is a great gateway to DS for computer scientists. You can start with dynamics on words and group actions on graphs; the basics of these field
by ykonstant 2y ago
Symbolic dynamics is a great gateway to DS for computer scientists. You can start with dynamics on words and group actions on graphs; the basics of these fields don't have too many prerequisites and have an immediate computational component you can exploit to augment your study with interesting code.
- DeathArrow 2y agoSounds interesting. Can you point to some problems in this field or some books or courses? Thanks!
- ykonstant 2y agoA fairly accessible intro (needs Linear Algebra) is Lind's book, An Introduction to Symbolic Dynamics and Coding. For dynamics on graphs, you may begin with any book on Spectral Graph Theory. A crucial theorem in symbolic dynamics is that entropy alone classifies Bernoulli systems (https://en.wikipedia.org/wiki/Bernoulli_scheme https://en.wikipedia.org/wiki/Bernoulli_scheme) up to isomorphism: if two Bernoulli systems have different entropies, they are structurally distinct. Starting from this fact, there are many problems trying to classify and distinguish symbolic systems based on computable invariants like entropy. Perhaps the most famous problem associated to symbolic dynamics is the x2 x3 conjecture. It says that the only probability distribution on [0,1) with 0 identified to 1 (i.e. the flat circle) that is invariant under both multiplication by 2 mod 1 and multiplication by 3 mod 1 is the uniform distribution. This was formulated by H. Furstenberg after he proved an analogous topological statement.