{"title":"Multiplicity of topological systems","authors":"DAVID BURGUET, RUXI SHI","doi":"10.1017/etds.2023.118","DOIUrl":null,"url":null,"abstract":"<p>We define the topological multiplicity of an invertible topological system <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(X,T)$</span></span></img></span></span> as the minimal number <span>k</span> of real continuous functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$f_1,\\ldots , f_k$</span></span></img></span></span> such that the functions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_i\\circ T^n$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$n\\in {\\mathbb {Z}}$</span></span></img></span></span>, <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240203084901549-0722:S0143385723001189:S0143385723001189_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$1\\leq i\\leq k,$</span></span></img></span></span> span a dense linear vector space in the space of real continuous functions on <span>X</span> endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/etds.2023.118","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We define the topological multiplicity of an invertible topological system $(X,T)$ as the minimal number k of real continuous functions $f_1,\ldots , f_k$ such that the functions $f_i\circ T^n$, $n\in {\mathbb {Z}}$, $1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.
$(X,T)$ as the minimal number k of real continuous functions 
$f_1,\ldots , f_k$ such that the functions 
$f_i\circ T^n$, 
$n\in {\mathbb {Z}}$, 
$1\leq i\leq k,$ span a dense linear vector space in the space of real continuous functions on X endowed with the supremum norm. We study some properties of topological systems with finite multiplicity. After giving some examples, we investigate the multiplicity of subshifts with linear growth complexity.
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