Open problems in symbolic dynamics
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, 69118,
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
"Nonexpansive 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 Zdsystems",
Invent. Math. 179 (2010), no. 1, 119158
(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: betashifts,
Sgap 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: Ktheory, 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