It follows immediately that X is hereditary iff r X “ X. Moreover, for any subshift p X, S q, one can definethe hereditary closure of X, p r X, S q, via r X : “ t z P t, u Z : z ď x for some x P X u. for basic propertiesand examples of such systems). We recall that subshift p X, S q with language L is hereditary if W P L, W ď W ñ W P L, here ď is to be understood coordinatewise (see e.g. t x P X : x r, | C | ´ s “ C u, for example, r s “ t x P X : x r s “ u. Given a block C P L p X q, we will often denote by the same letter C thecorresponding cylinder set, i.e. W “ t ď i ď n ´ w i “ u will be the number of ones in W., x j s for i ď j will denote a subword of x w j s for ď i ď j ď n ´ will denote a subword of W | W | “ n will stand for the length of W.Moreover, we will use words "block" and"word" interchangeably.Fix a word W “ r w w. When p X, S q is clearfrom the context we will sometimes abbreviate L “ L p X q, L n “ L n p X q, M “ M p X, S q and M e “ M e p X, S q. For any n ě, by L n p X q Ă L p X q we will denote the subset of blocks of length n. The set L p X q : “ Ť x P X L p x q is called the language of X. Recall that M p X, S q is compact (and metrizable) in the weak- ˚ topology.Let M e p X, S q Ă M p X, S q stand for the subset of ergodic measures.For x P X, let L p x q denote the family of all blocks appearing in x. Let M p X, S q be the set of probability Borel S -invariant measureson X. In this paper we consider subshifts p X, S q, where X Ă t, u Z and S denotesthe left shift. This is the conversetheorem to a recent result of Keller. Moreover, we show that B is taut whenever thecorresponding Mirsky measure ν η has full support. This answers an open ques-tion asked by Peckner. We show that the measure of maximal entropy for the hereditary clo-sure of a B -free subshift has the Gibbs property if and only if the Mirskymeasure of the subshift is purely atomic. A p r Hereditary subshifts whose measure of maximalentropy has no Gibbs property
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