Open Problems in Symbolic Dynamics

Last revised: 20 March 2016

This is a website devoted to presenting open problems in symbolic dynamics and tracking their solutions. I welcome suggestions for open problems, links to related problem sites and announcements of solved problems. The initial seed for the site is the survey
Open problems in symbolic dynamics ("OPSD"), from "Geometric and probabilistic structures in dynamics, 69--118, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008."

I'll post announcements of solutions to problems from OPSD. For those who solve problems stated in OPSD, I'd be grateful to hear about it, and for inclusion of OPSD in their references, so that future workers might easily find solutions in MathSciNet by tracking citations.

Problems from OPSD which have been solved:

• Problem 8.1 The most fundamental case is solved by Wolfgang Krieger in an arXiv preprint "On images of sofic systems". He gives necessary and sufficient conditions for existence of a factor map from a transitive sofic shift S onto a transitive aperiodic sofic shift T in the case that h(S)>h(T).
  
  
• Problem 11.1(i) What sets can be the set of expansive directions for a Z^2 SFT?
  This problem was solved by Pierre Guillon and Charalampos Zinoviadis. The solution, part of a larger work, is available on the Math arXiv and as part of the University of Turku PhD Thesis of Charalampos Zinoviadis.
  
  
• Questions 11.3 and 11.4 were answered in the affirmative by Mike Hochman in an ArXiv preprint "Non-expansive directions of Z^2 actions". . The final paper appeared in Ergodic Theory and Dynamical Systems.
  
  
• Problem 23.1 is solved and Question 23.2 answered by Giordano, Matui, Putnam and Skau in "Orbit equivalence for Cantor minimal Zd-systems", Invent. Math. 179 (2010), no. 1, 119-158 (and its arXiv preprint).
  
  
• Question 27.1 (and more) was answered in the affirmative by Mike Hochman in an Arxiv preprint "Isomorphism and embedding into Markov shifts off universally null sets". The final paper appeared in Acta Applicandae Mathematicae (2013) (Kim memorial volume).
  
  
• Problem 28.1 (due to Klaus Thomsen), "Must a subshift factor of a beta shift be intrinisically ergodic?" was answered in the affirmative by Vaughn Climenhaga and Daniel J. Thompson, in their ArXiv preprint
  Intrinsic ergodicity beyond specification: beta-shifts, S-gap shifts, and their factors. Their solution addresses much more than the beta shifts. The final paper appeared in Israel J.Math.
  
  
• Question 31.1 (Parry's finiteness question) (rephrased -- see the question for detail) Can there exist an infinite family of pairwise not topologically conjuagate group extensions by a finite abelian group G over a fixed irreducible shift of finite type, with the same periodic data?
  The answer is, drastically, yes. See
  Finite group extensions of shifts of finite type: K-theory, Parry and Livsic by M.Boyle and S.Schmieding
  
  
• Question 35.1 (Nivat's Conjecture) This was proved up to a factor of 2 by Van Cyr and Bryna Kra, who brought new ideas and proved other results, in their ArXiv preprint
  Nonexpansive Z^2 subdynamics and Nivat's conjecture
  For a proof of an interesting asymptotic version of the conjecture, see on the arXiv
  An Algebraic Geometric Approach to Nivat's Conjecture, by Jarkko Kari and Michal Szabados.
  
  

Related Open Problem Sites

• Mike Hochman's Open Problems Site (which has links to other open problems sites)
• Open Problems in Dynamical Systems and Ergodic Theory (and mirror site).
• Stony Brook Open Problems in Dynamical Systems