arXiv subjects

Neil Mañibo

Publications and source records attributed to Neil Mañibo.

16 recordsLinked to original sources

arXiv2026

Renormalisation techniques for inflation systems and some of their applications

Exact renormalisation techniques are important and powerful, particularly for inflation-generated systems. We review recent results in this direction. We recall the necessary notions for inflation systems and show the renormalisation principle, which allows us to obtain exact values of highly erratic functions, such as window covariograms. We apply these techniques to compute the diffraction pattern of the new monotile tilings with arbitrary precision. We also recall a recent invariant for system with pure-point spectrum, the orbit separation dimension, and its relation to renormalisation. Lastly, we recall results beyond the pure-point spectrum setting and show how renormalisation and Lyapunov exponents can be used to exclude the presence of absolutely continuous part of the spectra.

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arXiv2025

Radius-zero Extended Symmetries and Irregular Fibres of $\mathbb{Z}^d$-Substitution Subshifts

In this work, we consider $\mathbb{Z}^d$-shifts generated by digit substitutions. For such a shift $\mathbb{X}$, we study the elements of the normaliser of $\mathbb{Z}^d$ in the group of self homeomorphisms (called extended symmetries) whose local maps guaranteed by the generalised Curtis--Hedlund--Lyndon theorem have radius-zero. Using the formalism of minimal sets developed by Lemańczyk, Müllner and Yassawi, we provide an algorithm to compute elements of $\mathcal{N}(\mathbb{X})$ that preserve the hierarchical structure. We also investigate the interaction of extended symmetries with (i) the height lattice and (ii) the irregular fibres over the maximal equicontinuous factor. Towards (ii), we introduce the notion of derived substitutions to provide a complete description of the irregular fibres, extending a result by Coven, Quas and Yassawi in the one-dimensional case.

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arXiv2025

Substitutions on compact alphabets

We develop a systematic approach to continuous substitutions on compact Hausdorff alphabets. Focussing on implications of irreducibility and primitivity, we highlight important features of the topological dynamics of their (generalised) subshifts. We then reframe questions from ergodic theory in terms of spectral properties of a corresponding substitution operator. This requires an extension of standard Perron--Frobenius theory to the setting of Banach lattices. As an application, we identify computable criteria that guarantee quasi-compactness of the substitution operator. This allows unique ergodicity to be verified for several classes of examples. For instance, it follows that every primitive and constant length substitution on an alphabet with an isolated point is uniquely ergodic, a result which fails when there are no isolated points.

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arXiv2024

On the full centraliser of Erdős $\cB$-free shifts

The sets of $\cB$-free integers are considered with respect to (reversing) symmetries. It is well known that, for a large class of them, the centraliser of the associated $\cB$-free shift (otherwise known as its automorphism group) is trivial. We extend this result to the full centraliser, which effectively means to show that all self-homeomorphisms of the $\cB$-free shift that commute with some power of the shift are shifts themselves. This also leads to the result that the full normaliser agrees with the normaliser for this class, which is the semi-direct product of the centraliser with the cyclic group of order two generated by reflection.

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arXiv2024

Substitution tilings with transcendental inflation factor

For any $λ>2$, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor $λ$. In particular, we obtain the first class of examples of substitutive systems with transcendental inflation factors that possess usual dynamical properties enjoyed by primitive substitutions on finite alphabets. We show that both the associated subshift and tiling dynamical systems are strictly ergodic, which is related to the quasicompactness of the underlying substitution operator. We also provide an explicit substitution with transcendental inflation factor $λ$.

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arXiv2023

Substitutions and their generalisations

Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and potentials. In this review, we present some generalisations of substitutions, with a focus on substitutions on compact alphabets, and with an outlook towards their spectral theory. Guided by two main examples, we will illustrate what changes when one moves from finite to compact (infinite) alphabets, and discuss under which assumptions do we recover the usual geometric and statistical properties which make them viable models of materials with almost periodic order. We also present a planar example (which is a two-dimensional generalisation of the Thue--Morse substitution), whose diffraction is purely singular continuous.

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arXiv2023

Spectral theory of regular sequences: parametrisation and spectral characterisation

We extend the existence of ghost measures beyond nonnegative primitive regular sequences to a large class of nonnegative real-valued regular sequences. In the general case, where the ghost measure is not unique, we show that they can be parametrised by a compact abelian group. For a subclass of these measures, by replacing primitivity with a commutativity condition, we show that these measures have an infinite convolution structure similar to Bernoulli convolutions. Using this structure, we show that these ghost measures have pure spectral type. Further, we provide results towards a classification of the spectral type based on inequalities involving the spectral radius, joint spectral radius, and Lyapunov exponent of the underlying set of matrices. In the case that the underlying measure is pure point, we show that the support of the measure must be a subset of the rational numbers, a result that resolves a new case of the finiteness conjecture.

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arXiv2023

Torsion-free $S$-adic shifts and their spectrum

In this work we study $S$-adic shifts generated by sequences of morphisms that are constant-length. We call a sequence of constant-length morphisms torsion-free if any prime divisor of one of the lengths is a divisor of infinitely many of the lengths. We show that torsion-free directive sequences generate shifts that enjoy the property of quasi-recognizability which can be used as a substitute for recognizability. Indeed quasi-recognizable directive sequences can be replaced by a recognizable directive sequence. With this, we give a finer description of the spectrum of shifts generated by torsion-free sequences defined on a sequence of alphabets of bounded size, in terms of extensions of the notions of height and column number. We illustrate our results throughout with examples that explain the subtleties that can arise.

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arXiv2023

Catalan numbers as discrepancies for a family of substitutions on infinite alphabets

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.

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arXiv2023

Monochromatic arithmetic progressions in automatic sequences with group structure

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue--Morse and Rudin--Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A(d_n)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution.

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arXiv2022

Spectral properties of substitutions on compact alphabets

We consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non-constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

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arXiv2021

Spectral theory of spin substitutions

We introduce qubit substitutions in $\mathbb{Z}^m$, which have non-rectangular domains based on an endomorphism $Q$ of $\mathbb{Z}^m$ and a set $\mathcal{D}$ of coset representatives of $\mathbb{Z}^m/Q\mathbb{Z}^m$. We then focus on a specific family of qubit substitutions which we call spin substitutions, whose combinatorial definition requires a finite abelian group $G$ as its spin group. We investigate the spectral theory of the underlying subshift $(Σ,\mathbb{Z}^m)$. Under certain assumptions, we show that it is measure-theoretically isomorphic to a group extension of an $m$-dimensional odometer, which induces a complete decomposition of the function space $L^{2}(Σ,μ)$ . This enables one to use group characters in $\widehat{G}$ to derive substitutive factors and carry out a spectral analysis on specific subspaces. We provide general sufficient criteria for the existence of pure point, absolutely continuous and singular continuous spectral measures, together with some bounds on their spectral multiplicity.

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arXiv2021

Mixing properties and entropy bounds of a family of Pisot random substitutions

We consider a two-parameter family of random substitutions and show certain combinatorial and topological properties they satisfy. We establish that they admit recognisable words at every level. As a consequence, we get that the subshifts they define are not topologically mixing. We then show that they satisfy a weaker mixing property using a numeration system arising from a sequence of lengths of inflated words. Moreover, we provide explicit bounds for the corresponding topological entropy in terms of the defining parameters $n$ and $p$.

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arXiv2021

Ghost distributions of regular sequences are affine transformations of self-affine sets

Ghost measures of regular sequences---the unbounded analogue of automatic sequences---are generalisations of standard fractal mass distributions. They were introduced to determine fractal (or self-similar) properties of regular sequences similar to those related to automatic sequences. The existence and continuity of ghost measures for a large class of regular sequences was recently given by Coons, Evans and Mañibo. In this paper, we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions---the distribution functions of ghost measures---of the above-mentioned class of regular sequences are sections of self-affine sets. As an application of our result, we show that the ghost distributions of the Zaremba sequences---regular sequences of the denominators of the convergents of badly approximable numbers---are all singular continuous.

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arXiv2021

Admissible reversing and extended symmetries for bijective substitutions

In this paper, we deal with reversing and extended symmetries of shifts generated by bijective substitutions. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended symmetry, and algorithms how to check them. Moreover, we show that, for any finite group $G$ and any subgroup $P$ of the $d$-dimensional hyperoctahedral group, there is a bijective substitution which generates an aperiodic hull with symmetry group $\mathbb{Z}^{d}\times G$ and extended symmetry group $(\mathbb{Z}^{d} \rtimes P)\times G$.

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arXiv2017

Lyapunov exponents for binary substitutions of constant length

A method of confirming the absence of absolutely continuous diffraction via the positivity of Lyapunov exponents derived from the corresponding Fourier matrices is presented, which provides an approach that is independent of previous results on the basis of Dekking's criterion. This yields a positive result for all constant length substitutions on a binary alphabet which are primitive and aperiodic.

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