For any fixed alphabet A, the maximum topological entropy of a Z~d subshift with alphabet A is obviously log |A|. We study the class of nearest neighbor Z~d shifts of finite type (SFTs) which have topological entropy very close to this maximum, and show that they have many useful properties. Specifically, we prove that, for any d, there exists β_d such that, for any nearest neighbor Z~d SFT X with alphabet A for which (log |A|) -h(X) < β_d, X has a unique measure of maximal entropy μ. Our values of β_d decay polynomially (like O(d~(-17))) and we prove that the sequence must decay at least polynomially (like d~(-0.25+o(1))). We also show some other desirable properties for such X, for instance, that the topological entropy of X is computable and that μ is isomorphic to a Bernoulli measure. Although there are other sufficient conditions in the literature (see [Burton and Steif, Israel J. Math. 89 (1995) 275–300; H?ggstr?m, Israel J. Math. 94 (1996) 319–352; Markley and Paul, Lect. Notes Pure Appl. Math. 70 (1981) 135–157]) which guarantee a unique measure of maximal entropy for Z~d SFTs, this is (to our knowledge) the first such condition which makes no reference to the specific adjacency rules of individual letters of the alphabet.
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机译:对于任何固定的字母A,带有字母A的Z〜d子移位的最大拓扑熵显然是log | A |。我们研究了具有最接近该最大值的拓扑熵的有限类型最近邻Z〜d位移(SFT)的类别,并表明它们具有许多有用的特性。具体来说,我们证明,对于任何d,都存在β_d,从而对于字母(A)(log | A |)-h(X)<β_d的任何最近邻居Z〜d SFT X,X具有唯一的度量最大熵我们的β_d值呈多项式衰减(如O(d〜(-17))),并证明该序列必须至少呈多项式衰减(如d〜(-0.25 + o(1)))。我们还显示了此类X的其他一些理想属性,例如,X的拓扑熵是可计算的,而μ对贝努利测度而言是同构的。尽管文献中还有其他充分的条件(参见[Burton and Steif,Israel J. Math。89(1995)275-300; H?ggstr?m,Israel J. Math。94(1996)319-352; Markley和Paul,Lect。Notes Pure Appl。Math。70(1981)135–157])保证了Z〜d SFT的最大熵的唯一度量,这(据我们所知)是第一个没有提及单个字母的特定邻接规则。
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