{"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}
{"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":"35 5","pages":"Pages 914-930"},"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":"35 5","pages":"Pages 890-913"},"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":"35 5","pages":"Pages 837-864"},"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":"Correlation functions of the Rudin–Shapiro sequence","authors":"","doi":"10.1016/j.indag.2023.03.003","DOIUrl":"10.1016/j.indag.2023.03.003","url":null,"abstract":"<div><p><span>In this paper, we show that all odd-point correlation functions of the balanced Rudin–Shapiro sequence vanish and that all even-point correlation functions depend only on a single number, which holds for any weighted correlation function as well. For the four-point correlation functions, we provide a more detailed exposition which reveals some arithmetic structures and symmetries. In particular, we show that one can obtain the autocorrelation coefficients of its topological factor with maximal </span>pure point spectrum among them.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 771-795"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44470250","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":"35 5","pages":"Pages 959-988"},"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}
{"title":"A characterisation of linear repetitivity for cut and project sets with general polytopal windows","authors":"","doi":"10.1016/j.indag.2024.03.003","DOIUrl":"10.1016/j.indag.2024.03.003","url":null,"abstract":"<div><p>The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity (<strong>LR</strong>) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised <strong>LR</strong> for cut and project sets with convex polytopal windows whose supporting hyperplanes are commensurate with the lattice, the weak homogeneity property. For such cut and project sets, we show that <strong>LR</strong> is equivalent to two properties. One is a low complexity condition, which may be determined from the cut and project data by calculating the ranks of the intersections of the projection of the lattice to the internal space with the subspaces parallel to the supporting hyperplanes of the window. The second condition is that the projection of the lattice to the internal space is Diophantine (or ‘badly approximable’), which loosely speaking means that the lattice points in the total space stay far from the physical space, relative to their norm. We review then extend these results to non-convex and disconnected polytopal windows, as well as windows with polytopal partitions producing cut and project sets of labelled points. Moreover, we obtain a complete characterisation of <strong>LR</strong> in the fully general case, where weak homogeneity is not assumed. Here, the Diophantine property must be replaced with an inhomogeneous analogue. We show that cut and project schemes with internal space isomorphic to <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>⊕</mo><mi>G</mi><mo>⊕</mo><msup><mrow><mi>Z</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></math></span>, for <span><math><mi>G</mi></math></span> finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 1009-1056"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000156/pdfft?md5=2746cea224983adf4877799dace1bad2&pid=1-s2.0-S0019357724000156-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140203437","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":"Pure point diffraction and entropy beyond the Euclidean space","authors":"T. Hauser","doi":"10.1016/j.indag.2024.07.003","DOIUrl":"10.1016/j.indag.2024.07.003","url":null,"abstract":"<div><p>For Euclidean pure point diffractive Delone sets of finite local complexity and with uniform patch frequencies it is well known that the patch counting entropy computed along the closed centred balls is zero. We consider such sets in the setting of <span><math><mi>σ</mi></math></span>-compact locally compact Abelian groups and show that the topological entropy of the associated Delone dynamical system is zero. For this we provide a suitable version of the variational principle. We furthermore construct counterexamples, which show that the patch counting entropy of such sets can be non-zero in this context. Other counterexamples will show that the patch counting entropy of such a set cannot be computed along a limit and even be infinite in this setting.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 1057-1074"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357724000818/pdfft?md5=37da9342f84d0427094033cf2fe72940&pid=1-s2.0-S0019357724000818-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142121806","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":"Gap labels for zeros of the partition function of the 1D Ising model via the Schwartzman homomorphism","authors":"","doi":"10.1016/j.indag.2023.05.004","DOIUrl":"10.1016/j.indag.2023.05.004","url":null,"abstract":"<div><p>Inspired by the 1995 paper of Baake–Grimm–Pisani, we aim to explain the empirical observation that the distribution of Lee–Yang zeros corresponding to a one-dimensional Ising model<span> appears to follow the gap labelling theorem. This follows by combining two main ingredients: first, the relation between the transfer matrix formalism for the 1D Ising model and an ostensibly unrelated matrix formalism generating the Szegő recursion for orthogonal polynomials on the unit circle, and second, the gap labelling theorem for CMV matrices.</span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":"35 5","pages":"Pages 813-836"},"PeriodicalIF":0.5,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992589","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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