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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!

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    $\begingroup$ "noticeable": Maybe you meant "noteworthy"? $\endgroup$ Sep 30, 2021 at 17:32
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    $\begingroup$ or maybe "notable". $\endgroup$ Oct 1, 2021 at 7:42

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

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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

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