{"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":null,"pages":null},"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":null,"pages":null},"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":"Flow views and infinite interval exchange transformations for recognizable substitutions","authors":"","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":null,"pages":null},"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}
{"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":null,"pages":null},"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":null,"pages":null},"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":"Correlations of the Thue–Morse sequence","authors":"","doi":"10.1016/j.indag.2023.02.001","DOIUrl":"10.1016/j.indag.2023.02.001","url":null,"abstract":"<div><p>The pair correlations of the Thue–Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue–Morse sequence with general real weights are effectively determined by a single value of the balanced 2-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue–Morse sequence vanish, and that, for any even <span><math><mi>n</mi></math></span>, the <span><math><mi>n</mi></math></span>-point correlations of the balanced Thue–Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue–Morse system. We finish by showing how the correlations of the Thue–Morse system with general real weights can be derived from the balanced 2-point correlations.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43220423","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":"Catalan numbers as discrepancies for a family of substitutions on infinite alphabets","authors":"","doi":"10.1016/j.indag.2023.06.010","DOIUrl":"10.1016/j.indag.2023.06.010","url":null,"abstract":"<div><p><span>In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function is exponential (and guided by the inflation factor), the behaviour of the second-order term is strikingly different. For the finite setting, it is known that the second term is also exponential or exponential times a polynomial. We exhibit a large family of examples where the second term is at least exponential in </span><span><math><mi>n</mi></math></span> divided by half-integer powers of <span><math><mi>n</mi></math></span>, where <span><math><mi>n</mi></math></span><span> is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.</span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48677519","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":"Directional ergodicity, weak mixing and mixing for Zd- and Rd-actions","authors":"","doi":"10.1016/j.indag.2023.06.006","DOIUrl":"10.1016/j.indag.2023.06.006","url":null,"abstract":"<div><p>For a measure preserving <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>- or <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-action <span><math><mi>T</mi></math></span>, on a Lebesgue probability space <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></math></span>, and a linear subspace <span><math><mrow><mi>L</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span>, we define notions of direction <span><math><mi>L</mi></math></span> ergodicity, weak mixing, and strong mixing. For <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions, it is clear that these direction <span><math><mi>L</mi></math></span> properties should correspond to the same properties for the restriction of <span><math><mi>T</mi></math></span> to <span><math><mi>L</mi></math></span>. But since an arbitrary <span><math><mrow><mi>L</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></math></span> does not necessarily correspond to a nontrivial subgroup of <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, a different approach is needed for <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions. In this case, we define direction <span><math><mi>L</mi></math></span> ergodicity, weak mixing, and mixing in terms of the restriction of the unit suspension <span><math><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></math></span> to <span><math><mi>L</mi></math></span>, but also restricted to the subspace of <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>X</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>,</mo><mover><mrow><mi>μ</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>)</mo></mrow></mrow></math></span> perpendicular to the suspension direction. For <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions, we show (as is more or less clear for <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>) that these directional properties are spectral properties. For weak mixing <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>- and <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions, we show that directional ergodicity is equivalent to directional weak mixing. For ergodic <span><math><msup><mrow><mi>Z</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions <span><math><mi>T</mi></math></span>, we explore the relationship between direction <span><math><mi>L</mi></math></span> properties as defined via unit suspensions and embeddings of <span><math><mi>T</mi></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>-actions. Finally, ","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42215490","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":"The (reflected) Eberlein convolution of measures","authors":"","doi":"10.1016/j.indag.2023.10.005","DOIUrl":"10.1016/j.indag.2023.10.005","url":null,"abstract":"<div><p>In this paper, we study the properties of the Eberlein convolution of measures and introduce a reflected version of it. For functions we show that the reflected Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (reflected) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (reflected) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (reflected) Eberlein convolution.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136152169","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}
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