{"title":"Factorizations for quasi-Banach time–frequency spaces and Schatten classes","authors":"Divyang G. Bhimani , Joachim Toft","doi":"10.1016/j.indag.2024.09.005","DOIUrl":"10.1016/j.indag.2024.09.005","url":null,"abstract":"<div><div>We deduce factorization properties for Wiener amalgam spaces <span><math><mrow><mi>W</mi><mspace></mspace><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow></math></span>, an extended family of modulation spaces <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>ω</mi><mo>,</mo><mi>ℬ</mi><mo>)</mo></mrow></mrow></math></span>, and for Schatten symbols <span><math><msubsup><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>w</mi></mrow></msubsup></math></span> in pseudo-differential calculus under e.<!--> <!-->g. convolutions, twisted convolutions and symbolic products. Here <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>ω</mi><mo>,</mo><mi>ℬ</mi><mo>)</mo></mrow></mrow></math></span> can be any quasi-Banach Orlicz modulation space. For example we show that <span><math><mrow><mi>W</mi><mspace></mspace><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msup><mo>∗</mo><mi>W</mi><mspace></mspace><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>=</mo><mi>W</mi><mspace></mspace><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup></mrow></math></span> and <span><math><mrow><mi>W</mi><mspace></mspace><mspace></mspace><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>r</mi></mrow></msup><mi>#</mi><msubsup><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>w</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>w</mi></mrow></msubsup></mrow></math></span> when <span><math><mrow><mi>r</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, <span><math><mrow><mi>r</mi><mo>≤</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo><</mo><mi>∞</mi></mrow></math></span>. In particular we improve Rudin’s identity <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∗</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span>.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 838-879"},"PeriodicalIF":0.5,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-symmetric Jacobi polynomials of type BC1 as vector-valued polynomials, Part 1: Spherical functions","authors":"M. van Horssen, M. van Pruijssen","doi":"10.1016/j.indag.2024.09.003","DOIUrl":"10.1016/j.indag.2024.09.003","url":null,"abstract":"<div><div>We study non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> by means of vector-valued and matrix-valued orthogonal polynomials. The interpretation as matrix-valued orthogonal polynomials yields a new expression of the non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> in terms of the symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>. In this interpretation, the Cherednik operator, that has the non-symmetric Jacobi polynomials as eigenfunctions, corresponds to two shift operators for the symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>.</div><div>We show that the non-symmetric Jacobi polynomials of type <span><math><mrow><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> with so-called geometric root multiplicities, interpreted as vector-valued polynomials, can be identified with spherical functions on the sphere <span><math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>/</mo><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span> associated with the fundamental spin-representation of <span><math><mrow><mi>Spin</mi><mrow><mo>(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>. The Cherednik operator corresponds to the Dirac operator for the spinors on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span> in this interpretation.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 2","pages":"Pages 593-608"},"PeriodicalIF":0.5,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143463561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Herbert Batte , Mahadi Ddamulira , Juma Kasozi , Florian Luca
{"title":"Multiplicative independence in the sequence of k-generalized Lucas numbers","authors":"Herbert Batte , Mahadi Ddamulira , Juma Kasozi , Florian Luca","doi":"10.1016/j.indag.2024.09.002","DOIUrl":"10.1016/j.indag.2024.09.002","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mrow><mo>(</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></msubsup><mo>)</mo></mrow></mrow><mrow><mi>n</mi><mo>≥</mo><mn>2</mn><mo>−</mo><mi>k</mi></mrow></msub></math></span> be the sequence of <span><math><mi>k</mi></math></span>-generalized Lucas numbers for some fixed integer <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, whose first <span><math><mi>k</mi></math></span> terms are <span><math><mrow><mn>0</mn><mo>,</mo><mspace></mspace><mo>…</mo><mspace></mspace><mo>,</mo><mspace></mspace><mn>0</mn><mo>,</mo><mspace></mspace><mn>2</mn><mo>,</mo><mspace></mspace><mn>1</mn></mrow></math></span> and each term afterward is the sum of the preceding <span><math><mi>k</mi></math></span> terms. In this paper, we find all pairs of the <span><math><mi>k</mi></math></span>-generalized Lucas numbers that are multiplicatively dependent.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 819-837"},"PeriodicalIF":0.5,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143865104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Moebius maps which are characterized by the configuration of their dual maps","authors":"Fritz Schweiger","doi":"10.1016/j.indag.2024.09.001","DOIUrl":"10.1016/j.indag.2024.09.001","url":null,"abstract":"<div><div>Here we consider piecewise fractional linear maps with three branches. The paper presents a study of invariant measures with densities which can be written as infinite series. These series either have infinitely many poles or they sum up to a function with just one pole.</div></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 806-818"},"PeriodicalIF":0.5,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Projections of four corner Cantor set: Total self-similarity, spectrum and unique codings","authors":"Derong Kong , Beibei Sun","doi":"10.1016/j.indag.2024.08.006","DOIUrl":"10.1016/j.indag.2024.08.006","url":null,"abstract":"<div><div>Given <span><math><mrow><mi>ρ</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>/</mo><mn>4</mn><mo>]</mo></mrow></mrow></math></span>, the four corner Cantor set <span><math><mrow><mi>E</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span> is a self-similar set generated by the iterated function system <span><math><mrow><mrow><mo>{</mo><mrow><mo>(</mo><mi>ρ</mi><mi>x</mi><mo>,</mo><mi>ρ</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mi>ρ</mi><mi>x</mi><mo>,</mo><mi>ρ</mi><mi>y</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mi>ρ</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo>,</mo><mi>ρ</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mi>ρ</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo>,</mo><mi>ρ</mi><mi>y</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>ρ</mi><mo>)</mo></mrow><mo>}</mo></mrow><mo>.</mo></mrow></math></span> For <span><math><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></mrow></mrow></math></span> let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> be the orthogonal projection of <span><math><mi>E</mi></math></span> onto a line with an angle <span><math><mi>θ</mi></math></span> to the <span><math><mi>x</mi></math></span>-axis. In principle, <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> is a self-similar set having overlaps. In this paper we give a complete characterization on which the projection <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> is totally self-similar. We also study the spectrum of <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span>, which turns out that the spectrum achieves its maximum value if and only if <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> is totally self-similar. Furthermore, when <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> is totally self-similar, we calculate its Hausdorff dimension and study the subset <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>θ</mi></mrow></msub></math></span> which consists of all <span><math><mrow><mi>x</mi><mo>∈</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></math></span> having a unique coding. In particular, we show that <span><math><mrow><msub><mrow><mo>dim</mo></mrow><mrow><mi>H</mi></mrow></msub><msub><mrow><mi>U</mi></mrow><mrow><mi>θ</mi></mrow></msub><mo>=</mo><msub><mrow><mo>dim</mo></mrow><mrow><mi>H</mi></mrow></msub><msub><mrow><mi>E</mi></mrow><mrow><mi>θ</mi></mrow></msub></mrow></math></span> for Lebesgue almost every <span><math><mrow><mi>θ</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>)</mo></mrow></mrow></math></span>. Finally, for <span><math><mrow><mi>ρ</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>4</mn></mrow></ma","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"36 3","pages":"Pages 764-796"},"PeriodicalIF":0.5,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142180748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Critical curves of rotations","authors":"","doi":"10.1016/j.indag.2024.02.001","DOIUrl":"10.1016/j.indag.2024.02.001","url":null,"abstract":"<div><p>In rotations with a binary symbolic dynamics, a critical curve is the locus of parameters for which the boundaries of the partition that defines the symbolic dynamics are connected via a prescribed number of iterations and symbolic itinerary. We study the arithmetical and geometrical properties of these curves in parameter space.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 989-1008"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000089/pdfft?md5=2f9d5a6610f18fbac9a8c979bf5335b4&pid=1-s2.0-S0019357724000089-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139679261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Semicocycle discontinuities for substitutions and reverse-reading automata","authors":"","doi":"10.1016/j.indag.2023.05.003","DOIUrl":"10.1016/j.indag.2023.05.003","url":null,"abstract":"<div><p>In this article we define the semigroup associated to a primitive substitution. We use it to construct a minimal automaton which generates a substitution sequence <span><math><mi>u</mi></math></span> in reverse reading. We show, in the case where the substitution has a coincidence, that this automaton completely describes the <em>semicocycle discontinuities</em> of <span><math><mi>u</mi></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 796-812"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000496/pdfft?md5=0a7daca4bb4987075ce1de3e272a8290&pid=1-s2.0-S0019357723000496-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46621925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Automorphism groups of random substitution subshifts","authors":"","doi":"10.1016/j.indag.2023.08.006","DOIUrl":"10.1016/j.indag.2023.08.006","url":null,"abstract":"<div><p>We prove that for a suitably nice class of random substitutions, their corresponding subshifts have automorphism groups that contain an infinite simple subgroup and a copy of the automorphism group of a full shift. Hence, they are countable, non-amenable and non-residually finite. To show this, we introduce the concept of shuffles and generalised shuffles for random substitutions, as well as a local version of recognisability for random substitutions that will be of independent interest. Without recognisability, we need a more refined notion of recognisable words in order to understand their automorphisms. We show that the existence of a single recognisable word is often enough to embed the automorphism group of a full shift in the automorphism group of the random substitution subshift.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 931-958"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000836/pdfft?md5=3e4882926d47d4fbc572c02f04ab7895&pid=1-s2.0-S0019357723000836-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44793068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Inter-model sets in Rd are model sets","authors":"","doi":"10.1016/j.indag.2023.06.007","DOIUrl":"10.1016/j.indag.2023.06.007","url":null,"abstract":"<div><p>We show that any union of finitely many shifted model sets from a given cut-and-project scheme is a model set in some modified cut-and-project scheme. Restricting to direct space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, we show that any inter-model set is a model set in some modified cut-and-project scheme with second countable internal space. In both cases, the window in the modified cut-and-project scheme inherits the topological and measure-theoretic properties of the original windows.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 865-889"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44478350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Flow views and infinite interval exchange transformations for recognizable substitutions","authors":"Natalie Priebe Frank","doi":"10.1016/j.indag.2024.07.004","DOIUrl":"10.1016/j.indag.2024.07.004","url":null,"abstract":"<div><p>A flow view is the graph of a measurable conjugacy <span><math><mi>Φ</mi></math></span> between a substitution or S-adic subshift <span><math><mrow><mo>(</mo><mi>Σ</mi><mo>,</mo><mi>σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></math></span> and an exchange of infinitely many intervals in <span><math><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mi>F</mi><mo>,</mo><mi>m</mi><mo>)</mo></mrow></math></span>, where <span><math><mi>m</mi></math></span><span> is Lebesgue measure. A natural refining sequence of partitions of </span><span><math><mi>Σ</mi></math></span> is transferred to <span><math><mrow><mo>(</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mo>,</mo><mi>m</mi><mo>)</mo></mrow></math></span> using a canonical addressing scheme, a fixed dual substitution <span><math><msub><mrow><mi>S</mi></mrow><mrow><mo>∗</mo></mrow></msub></math></span>, and a shift-invariant probability measure <span><math><mi>μ</mi></math></span>. On the flow view, <span><math><mrow><mi>τ</mi><mo>∈</mo><mi>Σ</mi></mrow></math></span> is shown horizontally at a height of <span><math><mrow><mi>Φ</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow></math></span><span> using colored unit intervals to represent the letters.</span></p><p>The infinite interval exchange transformation <span><math><mi>F</mi></math></span> is well approximated by exchanges of finitely many intervals, making numeric and graphic methods possible. We prove that in certain cases a choice of dual substitution guarantees that <span><math><mi>Φ</mi></math></span> is self-similar. We discuss why the spectral type of <span><math><mrow><mi>Φ</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>Σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 1075-1103"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141773628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}