# Approximating Subshifts From Below

I'm looking to understand how to approximate certain countable alphabet subshifts by Markov shifts, and realised that I don't know how to do it even in the finite alphabet case. My guess is that the answer to the following question is well known, but I couldn't work it out this morning.

Let $X\subset\{0,1\}^{\mathbb Z}$ be closed and shift invariant. It is a classical result that there exist Markov shifts $X^n\supset X$ such that $$\lim_{n\to\infty} h_{top}(X^n)= h_{top}(X),$$ where

$$h_{top}(X):=\lim_{n\to\infty}\frac{1}{n}\log(\text{ no. of words of length n in X}).$$

This is easy to see, one just lets $X^n$ be the space made by freely concatenating all words of length $n$ in $X$, which is $n$-step Markov.

Question: Can we do such an approximation from below? i.e. Can we find Markov shifts $X_n\subset X$ with $\lim_{n\to\infty} h_{top} X_n= h_{top}(X)?$

## 3 Answers

If $X$ is minimal and not a periodic orbit then it cannot contain a periodic orbit and hence in particular cannot contain a Markov shift. A classical construction by Grillenberger shows that one can construct uniquely ergodic (hence in particular minimal) subshifts with arbitrary entropy. (This result also follows from the symbolic version of the Jewett-Krieger Theorem.)

• It should be Jewitt-Krieger. – MHS Nov 12 '14 at 1:36
• My spelling of "Jewett" was correct but I had indeed mis-spelled "Krieger". Thanks. – Ian Morris Nov 12 '14 at 11:11

Nevertheless there are approximations from below for other classes of subshifts, like sofic or coded systems. Depends on what your subshift looks like whether there is a sequence of proper subshifts approximating it in entropy from below or not.

• care to expand your answer? – Rodrigo A. Pérez Nov 12 '14 at 4:01
• What do you want me to explain in more detail? – MHS Nov 12 '14 at 13:49
• Thanks! In fact the article that Vaughn linked to is quite helpful for understanding when this can be done. – Tom Kempton Nov 14 '14 at 12:22

I guess that you can find such a statement in the first articles of Omri Sarig about thermodynamical formalism.

If your shift is mixing, then the pressure/entropy of your shift coincides with the supremum of pressure/entropy over all compact invariant subsets in restriction to which the shift is topologically mixing.

• Thanks Barbara, unfortunately my shift needn't be Markov, or even recodable as a countable Markov shift, so applying the results of Omri seems difficult. I've been trying to apply his methods, sometimes you can but sometimes not... – Tom Kempton Nov 17 '14 at 10:13