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Consider a subshift $X \subset \left\{0, \ldots, M \right\}^{\mathbb{N}}$. $X$ is said to be entropy-minimal if every subshift $Y \subsetneq X$ satisfies that $$h_{\mathrm{top}}(Y) < h_{\mathrm{top}}(X).$$ Equivalently, $X$ is entropy-minimal if for every word $\omega \in \mathcal{L}(X)$ the subshift $$ X_{\omega} = \left\{x \in X : \omega \text{ is not a factor of } x \right\}$$ satisfies that $h_{\mathrm{top}}(X_{\omega}) < h_{\mathrm{top}}(X)$.

It is a "folklore" result that irreducible subshifts of finite type of are entropy-minimal as well as subshifts with the specification property. Weaker notions of the specification property and entropy-minimality have been studied recently by Climenhaga, García-Ramos and Pavlov.

What I am looking for is a concrete example of a (one-dimensional) subshift $X$ that is non-entropy-minimal, transitive, and has positive topological entropy. It will be interesting if such example is a binary subshift.

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    $\begingroup$ Could one not take something like a positive entropy Toeplitz shift, the Thue-Morse shift, and then an extra 'generating element' which cycles through the language of both as you shift in either direction (in order to achieve transitivity). Then, I'm pretty sure the only proper subspaces which are subshifts are the two minimal components, and adding the TM shift and the generating element won't generate enough new words to increase the entropy, so the Toeplitz shift has the same entropy as the whole shift. $\endgroup$ – Dan Rust Feb 28 '20 at 9:10
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    $\begingroup$ Or in fact, far easier, forget the TM component. Just take a Toeplitz shift with entropy >0, then take any element of that shift and insert some new illegal word somewhere in the middle. This won't increase the entropy, by minimality of the Toeplitz shift, this new element will have a dense orbit. $\endgroup$ – Dan Rust Feb 28 '20 at 9:18
  • $\begingroup$ Dan, I will check what are you saying. I'm not familiar with Toeplitz subshifts. Fortunately, I found your work, so I will read it. Thanks $\endgroup$ – Rafael Alcaraz Barrera Feb 28 '20 at 18:01
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Let $f$ be a sublinear function that tends to infinity, such as $f(n) = \sqrt{n}$. Define $X \subset \{0,1,2\}^{\mathbb{N}}$ by forbidding all long enough words $w$ with more than $f(|w|)$ occurrences of $2$. Then $X$ has entropy $\log 2$ and is mixing, and properly contains the binary full shift, which likewise has entropy $\log 2$.

If you need a binary example, just replace the alphabet with three suitable binary words, like $01$, $001$ and $0001$.

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