arXiv subjects

Tony Samuel

Publications and source records attributed to Tony Samuel.

At least 19 recordsLinked to original sources

arXiv2026

Cut and project schemes in the Poincar\'e disc: From cocompact Fuchsian groups to chaotic Delone sets

A question raised by Davies et al [Phys. Rev. Lett. 131, 2023] is: "Can developing new cut and project models, where the lattice is not square or the curve is non-linear, generate better performing graded metamaterials?" In this article, we study a natural construction of such a cut and project scheme, namely, cut and project schemes in relation to cocompact Fuchsian groups acting on the Poincar\'e disc model of hyperbolic space. We present a condition on the fundamental domain (a hyperbolic polygon) of the group so that the resulting cut and project set $S \subset \mathbb{R}$ is a chaotic Delone set. We also investigate the set of tile lengths of $S$, namely $\mathcal{L}_{S} = \{ z - y : z,y \in S, \, z > y \; \text{and} \; (y,z) \cap S = \emptyset \}$, and show that this set is countably infinite. Finally, we apply our results to cocompact Fuchsian triangle groups and show that the resulting cut and project sets are chaotic Delone, complementing and extending the work of L\'{o}pez et al. [Discrete Contin. Dyn. Syst. 41, 2021].

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arXiv2026

On the geometry of generalised Koch snowflakes

We consider the geometry of a class of fractal sets in $\mathbb{R}^{2}$ that generalise the famous Koch curve and Koch snowflake. While the classical Koch curve is defined by an iterative process that divides a line segment into three parts and replaces the middle part by the legs of an isosceles triangle 'above' the line segment, in this more general setting, a choice can be made at each iteration as to whether to place this triangle 'above' or 'below' the line segment. The resulting fractals bear a striking visual resemblance to curves appearing in nature, such as coastlines and snowflakes. While these fractals can be generated by a random process that flips a coin each time to decide the orientation of the triangle, leading to 'almost sure' results for their geometrical properties, we define and study them deterministically to provide exact results. In particular, we show, using the theory of non-integer expansions, that the set of all possible values for the area enclosed by these generalised Koch curves is a closed interval. Moreover, we prove that the union of all these generalised snowflakes does not contain an open set, and has zero $2$-dimensional Lebesgue measure. Complementing these results, using arguments from calculus and fractal geometry, namely properties of geometric series and Frostman's Lemma, we show that each generalised Koch curve has infinite length and the same Hausdorff dimension as its classical counterpart. Further, we also give a classification for when a generalised Koch curve is a quasicircle.

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arXiv2025

The matrix potential game and structures of self-affine sets

We present a new variant of the potential game and show that certain compact subsets of $\mathbb{R}^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \mathbb{N}$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets.

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arXiv2025

The shadowing property for piecewise monotone interval maps

The property of shadowing has been shown to be fundamental in both the theory of symbolic dynamics as well as continuous dynamical systems. A quintessential class of discontinuous dynamical systems are those driven by transitive piecewise monotone interval maps and in particular $\beta$-transformations, namely transformations of the form $T_{\beta, \alpha} : x \mapsto \beta x + \alpha \; (\operatorname{mod} \, 1)$ acting on $[0,1]$. We provide a short elegant proof showing that this class of dynamical systems does not possess the property of shadowing, complementing and extending the work of Chen and Portela.

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arXiv2024

Rauzy fractals of random substitutions

We develop a theory of Rauzy fractals for random substitutions, which are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. We show that a Rauzy fractal can be associated with a given random substitution in a canonical manner, under natural assumptions on the random substitution. Further, we show the existence of a natural measure supported on the Rauzy fractal, which we call the Rauzy measure, that captures geometric and dynamical information. We provide several different constructions for the Rauzy fractal and Rauzy measure, which we show coincide, and ascertain various analytic, dynamical and geometric properties. While the Rauzy fractal is independent of the choice of (non-degenerate) probabilities assigned to a given random substitution, the Rauzy measure captures the explicit choice of probabilities. Moreover, Rauzy measures vary continuously with the choice of probabilities, thus provide a natural means of interpolating between Rauzy fractals of deterministic substitutions. Additionally, we highlight connections between Rauzy fractals and Rauzy measures of random substitutions and related S-adic systems.

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arXiv2022

Intermediate $\beta$-shifts as greedy $\beta$-shifts with a hole

We show that every intermediate $\beta$-transformation is topologically conjugate to a greedy $\beta$-transformation with a hole at zero, and provide a counterexample illustrating that the correspondence is not one-to-one. This characterisation is employed to (1) build a Krieger embedding theorem for intermediate $\beta$-transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [2019], and (2) obtain new metric and topological results on survivor sets of intermediate $\beta$-transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [2020]. Further, we derive a method to calculate the Hausdorff dimension of such survivor sets as well as results on certain bifurcation sets. Moreover, by taking unions of survivor sets of intermediate $\beta$-transformations one obtains an important class of sets arising in metric number theory, namely sets of badly approximable numbers in non-integer bases. We prove, under the assumption that the underlying symbolic space is of finite type, that these sets of badly approximable numbers are winning in the sense of Schmidt games, and hence have the countable intersection property, extending the results of Hu and Yu [2014], Tseng [2009] and F\"{a}rm, Persson and Schmeling [2010].

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arXiv2021

Measure theoretic entropy of random substitution subshifts

Subshifts of deterministic substitutions are ubiquitous objects in dynamical systems and aperiodic order (the mathematical theory of quasicrystals). Two of their most striking features are that they have low complexity (zero topological entropy) and are uniquely ergodic. Random substitutions are a generalisation of deterministic substitutions where the substituted image of a letter is determined by a Markov process. In stark contrast to their deterministic counterparts, subshifts of random substitutions often have positive topological entropy, and support uncountably many ergodic measures. The underlying Markov process singles out one of the ergodic measures, called the frequency measure. Here, we develop new techniques for computing and studying the entropy of these frequency measures. As an application of our results, we obtain closed form formulas for the entropy of frequency measures for a wide range of random substitution subshifts and show that in many cases there exists a frequency measure of maximal entropy. Further, for a class of random substitution subshifts, we prove that this measure is the unique measure of maximal entropy. These subshifts do not satisfy Bowen's specification property or the weaker specification property of Climenhaga and Thompson and hence provide an interesting new class of intrinsically ergodic subshifts.

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arXiv2020

Periodic intermediate $\beta$-expansions of Pisot numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are $\beta$-shifts, namely transformations of the form $T_{\beta, \alpha} \colon x \mapsto \beta x + \alpha \bmod{1}$ acting on $[-\alpha/(\beta - 1), (1-\alpha)/(\beta - 1)]$, where $(\beta, \alpha) \in \Delta$ is fixed and where $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1,2) \; \text{and} \; 0 \leq \alpha \leq 2-\beta \}$. Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045-2055, 2019), that the set of $(\beta, \alpha)$ such that $T_{\beta, \alpha}$ has the subshift of finite type property is dense in the parameter space $\Delta$. Here, they proposed the following question. Given a fixed $\beta \in (1, 2)$ which is the $n$-th root of a Perron number, does there exists a dense set of $\alpha$ in the fiber $\{\beta\} \times (0, 2- \beta)$, so that $T_{\beta, \alpha}$ has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the property of beginning sofic (that is a factor of a subshift of finite). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269-278, 1980) from the case when $\alpha = 0$ to the case when $\alpha \in (0, 2 - \beta)$. That is, we examine the structure of the set of eventually periodic points of $T_{\beta, \alpha}$ when $\beta$ is a Pisot number and when $\beta$ is the $n$-th root of a Pisot number.

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arXiv2019

Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

We consider the generalised Krein-Feller operator $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ with the natural restrictions that $\mu$ is atomless, the supp$(\nu)\subseteq$supp$(\mu)$ and the atoms of $\nu $ are embedded in the supp$(\mu)$. We show that the solutions of the eigenvalue problem for $\Delta_{\nu, \mu} $ can be transferred to the corresponding problem for the classical Krein-Feller operator $\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}$ with respect to the Lebesgue measure $\Lambda$ via an isometric isomorphism determined by the distribution function $F_\mu$ of $\mu$. In this way, we obtain a new characterisation of the upper spectral dimension and consolidate many known results on the spectral asymptotics of Krein-Feller operators. We also recover known properties of and connections to generalised gap diffusions associated to these operators.

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arXiv2018

On the continuity of entropy of Lorenz maps

We consider a one parameter family of Lorenz maps indexed by their point of discontinuity $p$ and constructed from a pair of bilipschitz functions. We prove that their topological entropies vary continuously as a function of $p$ and discuss Milnor's monotonicity conjecture in this setting.

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arXiv2018

Measure-geometric Laplacians on the real line

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\u{\i}n, given an atomless Borel probability measure $\eta$ supported on a compact subset of $\mathbb{R}$, Freiberg and Z\"{a}hle introduced a measure-geometric approach to define a first order differential operator $\nabla_{\eta}$ and a second order differential operator $\Delta_{\eta}$, with respect to $\eta$. We generalise this approach to measures of the form $\eta = \nu + \delta$, where $\nu$ is continuous and $\delta$ is finitely supported. We determine analytic properties of $\nabla_{\eta}$ and $\Delta_{\eta}$ and show that $\Delta_{\eta}$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $\Delta_{\eta}$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

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arXiv2018

Diffraction of return time measures

Letting $T$ denote an ergodic transformation of the unit interval and letting $f \colon [0,1)\to \mathbb{R}$ denote an observable, we construct the $f$-weighted return time measure $\mu_y$ for a reference point $y\in[0,1)$ as the weighted Dirac comb with support in $\mathbb{Z}$ and weights $f \circ T^z(y)$ at $z\in\mathbb{Z}$, and if $T$ is non-invertible, then we set the weights equal to zero for all $z < 0$. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of $\mu_{y}$ consists of a trivial atom and an absolutely continuous part, almost surely with respect to $y$. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations $T_{\alpha} \colon x \to x + \alpha \bmod{1}$ with rotation number $\alpha \in \mathbb{R}^+$. In contrast to when $T$ is mixing, we observe that the diffraction of $\mu_{y}$ is pure point, almost surely with respect to $y$. Moreover, if $\alpha$ is irrational and the observable $f$ is Riemann integrable, then the diffraction of $\mu_{y}$ is independent of $y$. Finally, for a converging sequence $(\alpha_{i})_{i \in \mathbb{N}}$ of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.

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arXiv2017

The Sierpi\'nski gasket as the Martin boundary of a non-isotropic Markov chain

In 2012 Lau and Ngai, motivated by the work of Denker and Sato, gave an example of an isotropic Markov chain on the set of finite words over a three letter alphabet, whose Martin boundary is homeomorphic to the Sierpi\'nski gasket. Here, we extend the results of Lau and Ngai to a class of non-isotropic Markov chains. We determine the Martin boundary and show that the minimal Martin boundary is a proper subset of the Martin boundary. In addition, we give a description of the set of harmonic functions.

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arXiv2017

On the density of intermediate \beta-shifts of finite type

We determine the structure of the set of intermediate $\beta$-shifts of finite type. Specifically, we show that this set is dense in the parameter space $\Delta = \{ (\beta, \alpha) \in \mathbb{R}^{2} \colon \beta \in (1, 2) \; \text{and} \; 0 \leq \alpha \leq 2 - \beta\}$. This generalises the classical result of Parry from 1960 for greedy and (normalised) lazy $\beta$-shifts.

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arXiv2017

Measure-geometric Laplacians for discrete distributions

In 2002 Freiberg and Z\"ahle introduced and developed a harmonic calculus for measure-geometric Laplacians associated to continuous distributions. We show their theory can be extended to encompass distributions with finite support and give a matrix representation for the resulting operators. In the case of a uniform discrete distribution we make use of this matrix representation to explicitly determine the eigenvalues and the eigenfunctions of the associated Laplacian.

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arXiv2016

Regularity of aperiodic minimal subshifts

At the turn of this century Durand, and Lagarias and Pleasants established that key features of minimal subshifts (and their higher-dimensional analogues) to be studied are linearly repetitive, repulsive and power free. Since then, generalisations and extensions of these features, namely $\alpha$-repetitive, $\alpha$-repulsive and $\alpha$-finite ($\alpha \geq 1$), have been introduced and studied. We establish the equivalence of $\alpha$-repulsive and $\alpha$-finite for general subshifts over finite alphabets. Further, we studied a family of aperiodic minimal subshifts stemming from Grigorchuk's infinite $2$-group $G$. In particular, we show that these subshifts provide examples that demonstrate $\alpha$-repulsive (and hence $\alpha$-finite) is not equivalent to $\alpha$-repetitive, for $\alpha > 1$. We also give necessary and sufficient conditions for these subshifts to be $\alpha$-repetitive, and $\alpha$-repulsive (and hence $\alpha$-finite). Moreover, we obtain an explicit formula for their complexity functions from which we deduce that they are uniquely ergodic.

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arXiv2016

A classification of aperiodic order via spectral metrics and Jarn\'ik sets

Given an $\alpha > 1$ and a $\theta$ with unbounded continued fraction entries, we characterise new relations between Sturmian subshifts with slope $\theta$ with respect to (i) an $\alpha$-H\"oder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\theta$, and (iii) complexity notions which we call $\alpha$-repetitive, $\alpha$-repulsive and $\alpha$-finite; generalisations of the properties known as linearly repetitive, repulsive and power free, respectively. We show that the level sets relate naturally to (exact) Jarn\'{\i}k sets and prove that their Hausdorff dimension is $2/(\alpha + 1)$.

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arXiv2014

A note on measure-geometric Laplacians

We consider the measure-geometric Laplacians $\Delta^{\mu}$ with respect to atomless compactly supported Borel probability measures $\mu$ as introduced by Freiberg and Z\"ahle in 2002 and show that the harmonic calculus of $\Delta^{\mu}$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $\Delta^{\mu}$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.

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