If $\mathcal {A}$ is a finite set (alphabet), the shift dynamical system consists of the space $\mathcal {A}^{\mathbb {N}}$ of sequences with entries in $\mathcal {A}$ , along with the left shift operator S. Closed S-invariant subsets are called subshifts and arise naturally as encodings of other systems. In this paper, we study the number of ergodic measures for transitive subshifts under a condition (‘regular bispecial condition’) on the possible extensions of words in the associated language. Our main result shows that under this condition, the subshift can support at most $({K+1})/{2}$ ergodic measures, where K is the limiting value of $p(n+1)-p(n)$ , and p is the complexity function of the language. As a consequence, we answer a question of Boshernitzan from 1984, providing a combinatorial proof for the bound on the number of ergodic measures for interval exchange transformations.

中文翻译：

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常规双特条件下传递子移位的遍历测度数

如果$\数学{A}$是一个有限集（字母表），移位动力系统由空间组成$\mathcal {A}^{\mathbb {N}}$具有条目的序列$\数学{A}$, 以及左移运算符小号. 关闭小号- 不变的子集被称为子移位，并且作为其他系统的编码自然而然地出现。在本文中，我们研究了在相关语言中可能的单词扩展的条件（“常规双特殊条件”）下传递子移位的遍历度量的数量。我们的主要结果表明，在这种情况下，subshift最多可以支持$({K+1})/{2}$遍历措施，其中ķ是极限值$p(n+1)-p(n)$， 和p是语言的复杂度函数。因此，我们从 1984 年开始回答 Boshernitzan 的问题，为区间交换变换的遍历度量数量的界限提供了组合证明。