Perhaps the biggest open problem in symbolic dynamics is the equivalence problem for subshifts of finite type.
Given a finite alphabet $\mathcal{A}$ and a finite set $\mathcal{F}$ of finite words over $\mathcal{A}$ (the forbidden words), the corresponding subshift of finite type consists of
The space $S\subseteq \mathcal{A}^\mathbb{Z}$ of all bi-infinite words over $\mathcal{A}$ that do not have any of the forbidden words as subwords, and
The shift map $\sigma \colon S\to S$ that shifts each symbol to the left one spot.
Two subshifts $(S,\sigma)$ and $(S',\sigma')$ of finite type are equivalent if they are conjugate as dynamical systems, i.e. if there exists a homeomorphism $h\colon S\to S'$ such that $h\circ \sigma = \sigma'\circ h$.
Is there an algorithm to determine whether two subshifts of finite type $(S,\sigma)$ and $(S',\sigma')$ are equivalent?
See M. Boyle, Open problems in symbolic dynamics. Contemporary mathematics 469 (2008): 69-118.
A solution to this problem was famously published in the Annals of Mathematics by R. F. Williams in a 1973 paper. An error was found in his proof, so the correctness of his main classification algorithm became the "Williams conjecture". This conjecture was disproven by K. H. Kim and F. W. Roush in 1990's in a series of two papers, and at present we have essentially no idea whether equivalence is decidable.