# Neil MañiboThe Open University (UK) · Department of Mathematics and Statistics

Neil Mañibo

Dr. math.

## About

26

Publications

603

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86

Citations

Citations since 2017

Introduction

**Skills and Expertise**

## Publications

Publications (26)

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 striki...

Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba’s conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of s...

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-rec...

For any $\lambda>2$, we construct a substitution on an infinite alphabet which gives rise to a substitution tiling with inflation factor $\lambda$. In particular, we obtain the first class of examples of substitutive systems with transcendental inflation factors. We also show that both the associated subshift and tiling dynamical systems are strict...

In this paper, we deal with reversing and extended symmetries of subshifts generated by bijective substitutions. We survey some general algebraic and dynamical properties of these subshifts and recall known results regarding their symmetry groups. We provide equivalent conditions for a permutation on the alphabet to generate a reversing/extended sy...

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 numer...

We develop a general theory of continuous substitutions on compact Hausdorff alphabets. Focussing on implications of primitivity, we provide a self-contained introduction to the topological dynamics of their subshifts. We then reframe questions from ergodic theory in terms of spectral properties of the corresponding substitution operator. The stand...

We introduce substitutions in \begin{document}$ {\mathbb{Z}}^m $\end{document} which have non-rectangular domains based on an endomorphism \begin{document}$ Q $\end{document} of \begin{document}$ {\mathbb{Z}}^m $\end{document} and a set \begin{document}$ {\mathcal D} $\end{document} of coset representatives of \begin{document}$ {\mathbb{Z}}^m/Q{\ma...

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...

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 clas...

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...

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 numer...

In the study of spectral properties of a d-dimensional aperiodic tiling which arises from an inflation rule ρ on a finite set of prototiles.

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 hyp...

Regular sequences are natural generalisations of fixed points of constant-length substitutions on finite alphabets, that is, of automatic sequences. Indeed, a regular sequence that takes only finitely many values is automatic. Given a $k$-regular sequence $f$, there is an associated vector space $\mathcal{V}_k(f)$, which is analogous to the substit...

We show that the Mahler measure of every Borwein polynomial—a polynomial with coefficients in \( \{-1,0,1 \}\) having non-zero constant term—can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer’s problem for height-one polynomials having m...

We use generalised Zeckendorf representations of natural numbers to investigate mixing properties of symbolic dynamical systems. The systems we consider consist of bi-infinite sequences associated with so-called random substitutions. We focus on random substitutions associated with the Fibonacci, tribonacci and metallic mean numbers and take advant...

The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diff...

The family of primitive binary substitutions defined by \(1 \mapsto 0 \mapsto 0 1^m\) with \(m\in \mathbb {N}\) is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation (\(m=1\)), the inflation ru...

The pair correlations of primitive inflation rules are analysed via their exact renormalisation relations. We introduce the inflation displacement algebra that is generated by the Fourier matrix of the inflation and deduce various consequences of its structure. Moreover, we derive a sufficient criterion for the absence of absolutely continuous diff...

We show that the Mahler measure of every Borwein polynomial --- a polynomial with coefficients in $ \{-1,0,1 \}$ having non-zero constant term --- can be expressed as a maximal Lyapunov exponent of a matrix cocycle that arises in the spectral theory of binary constant-length substitutions. In this way, Lehmer's problem for height-one polynomials ha...

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...