A 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 an

A 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.