Indagationes Mathematicae-New Series最新文献

A flow view is the graph of a measurable conjugacy $\Phi$ between a substitution or S-adic subshift $(\Sigma ,\sigma ,\mu )$ and an exchange of infinitely many intervals in $([0,1],F,m)$, where $m$ is Lebesgue measure. A natural refining sequence of partitions of $\Sigma$ is transferred to $([0,1],m)$ using a canonical addressing scheme, a fixed dual substitution $S_{\ast }$, and a shift-invariant probability measure $\mu$. On the flow view, $\tau \in \Sigma$ is shown horizontally at a height of $\Phi \left(\tau \right)$ using colored unit intervals to represent the letters.

The infinite interval exchange transformation $F$ 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 $\Phi$ is self-similar. We discuss why the spectral type of $\Phi \in L^2(\Sigma ,\mu ),$ is of particular interest. As an example of utility, some spectral results for constant-length substitutions are included.

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 $n$, the $n$-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.

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 $n$ divided by half-integer powers of $n$, where $n$ is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.

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 pure point spectrum among them.

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.

The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity (LR) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised LR 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 LR 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 LR 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 $R^n\oplus G\oplus Z^r$, for $G$ finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space $R^n$, so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.

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 $\sigma$-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.

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 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.