On metric properties of substitutions
On metric properties of substitutions
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关于替代的度量性质
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发表时间：
1988
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通讯作者：
M. K. Mentzen
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作者：
M. Lemanczyk;M. K. Mentzen
发展了恒定长度替换的度量理论。给出了此类离散频谱的一些特征。考察了一些直接产品中局部排名第一的现象。确定了双射替换的测度论中心化子。简介 令r 2 为自然数。我们将用 Nr 表示集合 {0, 1, ..., r 1} 。设 N*r = ~n1 N" 。 N*r 的元素称为块。如果 B E N,* , B = (bo... bn-1) 则 B[s, t] = (b,... b, ), B[s, s] = B[s]，n 称为 B 的长度, )B ) = n。这些符号可以以明显的方式扩展到 N,7L 的元素。令 À a 2 和 0 ： Nr ~ Nr 。存在从 Nk 到 Nk03BBr 的映射以及从 N,7L 到自身的映射（也用 0 表示）的自然扩展，如下所示。我们用 0" 表示 0 的 n 次组合。如果存在 n a 1 使得对于每个 i, j E Nr ， (1) 03B8n(i)[k] = j ，则这样的映射称为 r 符号上的恒定长度替换一些 k k 取决于 i, j)。众所周知，Nr 中存在元素 i 和 j，使得对于某些 p，0P(j) 的第一个符号是 j，03B8p(i) 的最后一个符号是 i。我们将 xo E N,7L 定义如下： 242 那么 xo 是固定点 0。令 T 为 NZr 上的移位。设 X(03B8) = Tn(x0): n ~ Z} 表示 xo 轨道的闭合。则 (X(0), T) 是唯一遍历动力系统。我们用 J1 表示唯一的 T 不变测度。我们假设 0 的替换是非循环的，即 X(0) 是一个无限集。评论。读者应该能够在参考文献中列出的论文中找到一些在论文中未定义的有关遍历理论的术语。本文关注恒定长度替换类的两个测度理论不变量：秩和集中器。在[11]中，第一作者证明了对于两个符号的替换，等级表征了离散谱。在本文中，我们证明以下内容： 定理 1。如果 0 是恒定长度的替换，则替换 (X(03B8), T, 03BC) 具有离散谱，当且仅当 T 的秩为 1。离散谱的另一个特征给出了定理 2。替换 (X(03B8), T, 03BC) 具有离散谱，当且仅当 T 是刚性的。在[2]中，Ferenczi 引入了局部一级属性的概念。由于每个有限秩自同构具有局部秩 1，因此下面的定理 3 表明用任何非周期自同构替换恒定长度（具有部分连续谱）的每个遍历乘积都不是有限秩变换。定理 3. 令 (X(03B8), T, 03BC) 为具有部分连续谱的恒定长度替代，并令 1": (Z, m) ~ (Z, m) 为任意非周期自同构，使得 T x T 遍历。则 T x 03C4 不是局部秩 1。推论 4. 令 T’ 为局部秩 1 变换。假设 T 是替代，使得 03C4 和 T 是因子如果 T x 03C4 是遍历的，则 03C4 是有限群上的旋转，因此 T x 03C4 是 T' 的一个因子，因此它必须具有局部秩 1 性质。在 [8] 中，J. King 引入了本质的概念。 243 中心化器 EC(T) = C(T)/{Tn} 在[10]中提出了每个有限群是否被实现为某些自同构的本质中心化器的问题，从[8]可以看出，等级为 1 的自同构是不可能的。为了解决这个问题，我们引入了一些定义和符号。在[15]之后，我们说 r 符号上的恒定长度 03BB 的替换 0 是双射的，如果 其中d(b1b2...bk, c1c2... Ck) = 卡 {i: b, ~ ci}/k。如果是这种情况，那么我们可以用集合 Nr 的排列 {03B80..... 03B803BB-1} 来识别矩阵 03B8(i)[t] 中的列，其中 i 03B8(i)[t]。由 {03B80, ... , 03B803BB-1} 生成的传递性地作用于 Nr。 不失一般性，我们可以假设 03B80 = id。让我们用 C(0) 表示 Sr 中的 G(03B8) 的中心化器。现在我们可以得出以下结果： 定理 5. 令 03B8 为双射替换，然后令 G 为有限群。对于某些 r > 1，G(03C31, ... , a,,) 与某个集合 {03C31, 03C32, ..., 03C303BB} c Sr 同构，这样就足以证明 G(03C31, ... , a,,) 传递作用于 Nr。让我们注意到，如果 G(03C31, ... , 03C303BB) 传递作用于 Nr，则C(G(03C31, ... , 03C303BB)) c Sr 最多有 r 个元素。现在考虑 9, 03C8: G ~ Sr，由以下公式给出： 很明显，03C8(G) 传递作用于 Nr 且 C(03C8(G)) = ~(G) ~ G。 推论 6. 存在非弱混合的自同构（特别是非弱混合的自同构）。素数），具有平凡的集中器。作为一个问题，我们提出了具有部分连续谱的遍历自同构 T 的类型的有限扩展。我们从与常数长度替换的概念相关的一些定义开始。 r 符号上的 03BB，并令 xo 为由 (2) 定义的固定点 0。 h(0) = max (n a 1: g.c.d. (n, 03BB) = 1, n 除 g.c.d. {t:x0[t] ~ x0[0]}} 将称为 0 的高度 ([1], p. 226)。如果 h(O) = 1，则代入 0 称为纯代入（[1]，第 229 页）如果 0 不是纯的，则存在恒定长度 03BB 的纯替换 ~，使得
The metric theory of substitutions of constant length is developed. Some characterizations of discrete spectrum in this class are given. Local rank one phenomen in a class of some direct produts is examined. The measure-theoretic centralizer of bijective substitutions is determined. Introduction Let r 2 be a natural number. We will denote the set {0, 1, ..., r 1} by Nr . Let N*r = ~n1 N" . The elements of N*r are called blocks. If B E N,* , B = (bo... bn-1) then B[s, t] = (b,... b, ), B[s, s] = B[s], and n is called the length of B, )B ) = n. These notations can be extended to the elements of N,7L in an obvious way. Let À a 2 and 0 : Nr ~ Nr . There is a natural extension of 0 to a map from Nk into Nk03BBr and to a map from N,7L into itself (denoted also by 0) given as follows We denote by 0" the n-fold composition of 0. Such a map is called a substitution of constant length on r symbols if there exists n a 1 such that for each i, j E Nr , (1) 03B8n(i)[k] = j for some k k depends on i, j). It is well-known that in Nr there exist elements i and j such that for some p the first symbol of 0P( j) is j and the last symbol of 03B8p(i) is i. We define xo E N,7L as follows: 242 Then xo is a fixed point of 0. Let T be the shift on NZr. Let X(03B8) = Tn(x0): n ~ Z} denote the closure of the orbit of xo. Then (X(0), T) is a uniquely ergodic dynamical system. We denote the unique T-invariant measure by J1. We will assume that the substitution regarded 0 is noncyclic i.e., X(0) is an infinite set. REMARK. The reader should be able to find some of the terms concerning ergodic theory which are undefined in the paper in the papers listed in the References. This paper is concerned with two measure-theoretic invariants for the class of substitutions of constant length: the rank and the centralizer. In [11] the first author has proved that for the substitutions on two symbols the rank characterizes the discrete spectrum. In this paper we prove the following: THEOREM 1. If 0 is a substitution of constant length then the substitution (X(03B8), T, 03BC) has discrete spectrum iff T has rank 1. Another characterization of discrete spectrum gives THEOREM 2. The substitution (X(03B8), T, 03BC) has discrete spectrum iff T is rigid. In [2], Ferenczi introduced the notion of the local rank 1 property. Since each finite rank automorphism has local rank 1, Theorem 3 below shows that each ergodic product of a substitution of constant length (with partly continuous spectrum) with any aperiodic automorphism is not a finite rank transformation. THEOREM 3. Let (X(03B8), T, 03BC) be a substitution of constant length with partly continuous spectrum and let 1": (Z, m) ~ (Z, m) be any aperiodic automorphism such that T x T is ergodic. Then T x 03C4 is not local rank 1. COROLLARY 4. Let T’ be a local rank 1 transformation. Suppose T is substitution such that 03C4 and T are factors of T’. If T x 03C4 is ergodic then 03C4 is a rotation on a finite group. The corollary follows from the fact that T is disjoint from 03C4. Thus T x 03C4 is a factor of T’ and hence it must have the local rank 1 property. The second invariant considered in this paper is the measure-theoretic centralizer, C(T). In [8] J. King introduced the notion of the essential 243 centralizer EC(T) = C(T)/{Tn}. In [10] the question of whether each finite group is realized as the essential centralizer of some automorphism has been raised. From [8] it follows that it is not possible for automorphisms with rank 1. To solve this problem we introduce some definitions and notations. Following [15] we say that substitution 0 of constant length 03BB on r symbols is bijective if where d(b1b2...bk, c1c2... Ck) = card {i: b, ~ ci}/k. If this is the case, then we may identify the columns in the matrix 03B8(i)[t] with the permutations {03B80..... 03B803BB-1} of the set Nr, where i 03B8(i)[t]. Observe, that the group G(03B8) c Sr generated by {03B80, ... , 03B803BB-1} acts transitively on Nr. Without loss of generality we may assume that 03B80 = id. Let us denote by C(0) the centralizer of G(03B8) in Sr. Now we can formulate the following results: THEOREM 5. Let 03B8 be a bijective substitution. Then Now, let G be a finite group. To realize G as the essential centralizer of some bijective substitution it is sufficient to show that G is isomorphic to the centralizer of some set {03C31, 03C32, ..., 03C303BB} c Sr for some r > 1, such that G(03C31, ... , a,,) acts transitively on Nr. Let us notice that if G(03C31, ... , 03C303BB) acts transitively on Nr then C(G(03C31, ... , 03C303BB)) c Sr has at most r elements. Consider now 9, 03C8: G ~ Sr given by the formulas: It is clear that 03C8(G) acts transitively on Nr and C(03C8(G)) = ~(G) ~ G. COROLLARY 6. There exists an automorphism which is not weakly-mixing (in particular which is not prime), with trivial centralizer. COROLLARY 7. There exists a finite extension of some dynamical system with discrete spectrum which has trivial centralizer. As a problem, we state the question of what kinds of groups are realized as the EC(T) for some ergodic automorphism T with partly continuous spectrum. For further discussion we refer to the last section. 244 Proofs. We start with some definitions connected with the notion of substitution of constant length. Let 0 be a substitution of constant length 03BB on r symbols, and let xo be a fixed point of 0 defined by (2). The number h(0) = max (n a 1: g.c.d. (n, 03BB) = 1, n divides g.c.d. {t:x0[t] ~ x0[0]}} will be called the height of0 ([1], p. 226). Substitution 0 is called pure if h(O) = 1 ([1], p. 229). If0is not pure then there is a pure substitution ~ of constant length 03BB such that