Automatic sequences are orthogonal to aperiodic multiplicative functions

M. Lema'nczyk, C. Mullner
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引用次数: 15

Abstract

Given a finite alphabet $\mathbb{A}$ and a primitive substitution $\theta:\mathbb{A}\to\mathbb{A}^\lambda$ (of constant length $\lambda$), let $(X_\theta,S)$ denote the corresponding dynamical system, where $X_{\theta}$ is the closure of the orbit via the left shift $S$ of a fixed point of the natural extension of $\theta$ to a self-map of $\mathbb{A}^{\mathbb{Z}}$. The main result of the paper is that all continuous observables in $X_{\theta}$ are orthogonal to any bounded, aperiodic, multiplicative function $\mathbf{u}:\mathbb{N}\to\mathbb{C}$, i.e. \[ \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(S^nx)\mathbf{u}(n)=0\] for all $f\in C(X_{\theta})$ and $x\in X_{\theta}$. In particular, each primitive automatic sequence, that is, a sequence read by a primitive finite automaton, is orthogonal to any bounded, aperiodic, multiplicative function.
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自动序列与非周期乘法函数是正交的
给定一个有限字母$\mathbb{A}$和一个原始替换$\theta:\mathbb{A}\to\mathbb{A}^\lambda$(长度恒定$\lambda$),让$(X_\theta,S)$表示相应的动力系统,其中$X_{\theta}$是轨道的闭包,通过将$\theta$的自然扩展的不动点向左移动$S$到$\mathbb{A}^{\mathbb{Z}}$的自映射。本文的主要结果是$X_{\theta}$中所有的连续可观测值与任何有界的、非周期的、乘法函数$\mathbf{u}:\mathbb{N}\to\mathbb{C}$正交,即对于所有的$f\in C(X_{\theta})$和$x\in X_{\theta}$都是\[ \lim_{N\to\infty}\frac1N\sum_{n\leq N}f(S^nx)\mathbf{u}(n)=0\]。特别地,每个原始自动序列,即由原始有限自动机读取的序列,与任何有界的、非周期的、乘法函数正交。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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