TITLE: Open problems in symbolic dynamics
QUESTION [14 upvotes]: I would like to know  which are some noticeable open problems in symbolic dynamics, including substitution dynamics. I'm especially interested in connections with topological chaos of various forms. Thanks!

REPLY [7 votes]: You mentioned substitution systems, so the Pisot substitution conjecture obviously has to be mentioned. There's a (mostly) up to date exposition by Akiyama, Barge, Berthé, Lee and Siegel that can be read here.
In a nutshell, it asks whether it is true that an irreducible Pisot substitution on a finite alphabet always gives rise to a subshift which has pure point dynamical spectrum.
The conjecture is known to be true for all two-letter substitutions, but probably the most interesting case that has been solved is by Barge, where he solved the conjecture for all `$\beta$-substitutions' - https://arxiv.org/abs/1505.04408