Subshifts with the same entropy - MathOverflow

Subshifts with the same entropy

2011-04-30T23:43:24Z

It is known that two Markov subshifts with the same entropy are "almost isomorphic" (up to a subset of measure 0) if the entropy is a logarithm of an integer (see R. L. Adler, L. W. Goodwyn, and B.Weiss. Equivalence of topological markov shifts. Israel J. Math, 27(1):49--63, 1977). Is it true (known) if the entropy is not a logarithm of an integer?

Update Basically I want to know if the result of AGW is true without the integer assumption. I would appreciate an answer of the form "yes plus reference" or "no plus reference" or "it is still unknown".

Answer by Nishant Chandgotia for Subshifts with the same entropy

2011-05-01T00:15:36Z

Look at Section 9 of http://www-users.math.umd.edu/users/mmb/papers/openfinalsub3nov2007.pdf

Answer by Mark Sapir for Subshifts with the same entropy

2011-05-01T10:16:21Z

Here is an answer of Benjy Weiss. This answers my question completely:

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The general case is treated in:

Adler, Roy L.; Marcus, Brian

Topological entropy and equivalence of dynamical systems.

Mem. Amer. Math. Soc. 20 (1979), no. 219, iv+84 pp.

The result is certainly described in the book by Lind and Marcus on Symbolic Dynamics and if I remember correctly they also give an exposition of the proof.

Answer by Stephen Shea for Subshifts with the same entropy

2011-05-01T12:58:52Z

Although you have answered your own question completely, I'd like to add a reference to a closely related open question from Mike Hochman. I believe readers interested in almost isomorphisms will also be interested in this open question. One can consider two equal entropy mixing shifts of finite type, $X$ and $Y$, that are not topologically conjugate. Let $Per(X)$ and $Per(Y)$ be the periodic points of $X$ and $Y$ respectively. Are $X$ \$Per(X)$ and $Y$ \$Per(Y)$ topologically conjugate?

Here is a link with a formal description of the problem and some remarks.

From Open Problems section of the 3rd Pingree Park Workshop (2010):

http://www.math.princeton.edu/%7Ehochman/open-problems/pingree-open-problems.pdf

I'd also like to add that there is a closely related theory of finitary isomorphisms of Markov processes (Keane and Smorodinsky for finite state, Rudolph for countable state) that the reader may be interested in.

Answer by Marcelo for Subshifts with the same entropy

2011-06-26T00:19:32Z

The main theorem in Adler, Roy L.; Marcus, Brian

"Topological entropy and equivalence of dynamical systems"

doesn't use the fact that the entropy is log of a natural number. However it states that two ergodically supported topological Markov shifts are almost topologically conjugated if and only if them have the same topological entropy and the same ergodic period.

There is a work of Wenxiang Sun which extends this result for general expansive, ergodically supprted maps that have the shadowing property:

http://iopscience.iop.org/0951-7715/13/3/309