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!

## 2 Answers

Mike Boyle once compiled a pretty large collection of open problems in symbolic dynamics, and has been keeping track of their status.

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