{"title":"Automatic sequences are orthogonal to aperiodic multiplicative functions","authors":"M. Lema'nczyk, C. Mullner","doi":"10.3934/dcds.2020260","DOIUrl":null,"url":null,"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.","PeriodicalId":335105,"journal":{"name":"2019-20 MATRIX Annals","volume":"35 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"15","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"2019-20 MATRIX Annals","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/dcds.2020260","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.