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.

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.

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 β-transformations, namely transformations of the form T_β, α : x ↦β x + α (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.

In engineering, the design of a product relies heavily on a design specification; a co-creation of customer and engineer which captures the requirements. Subjectivity is intrinsic to this process. Whilst engineers typically have a high appreciation of the technical aspects of design, the detailed knowledge of environmental and socioeconomic (ESE) implications are often held elsewhere. As such, efficient and effective design is critically dependent on the processes underpinning knowledge transfer. However, the information interfaces between engineering and the requirements of our swiftly changing civilization remain indirect and suboptimal, and the unintended consequences of design choices are becoming increasingly serious. Transdisciplinary engineering bridges knowledge boundaries interfacing with engineering (e.g. social science). This paper explores whether topology (a branch of pure mathematics) presents an opportunity to analyse the complex interdependency of transdisciplinary engineering information. Topology and geometry describe the structure of objects such as connectedness or the number of holes and have recently provided a suite of powerful and robust tools for analysing high-dimensional data sets. However, the real-world implementation of the term topology is still evolving. Interviews with engineering organisations, revealed that topology is predominately interpreted as ‘Topology Optimisation’ in the context of advanced design and manufacturing. To date, mathematical processes for critically and systematically examining the topology of systems have not been transferred through to the engineering industry. This paper contrasts how topology is interpreted by the engineering industry compared to academic literature and reflects on the opportunities of applying the mathematical theory of topological analysis to transdisciplinary engineering data.

We show that every intermediate β -transformation is topologically conjugate to a greedy β -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 β -transformation, complementing the result of Li, Sahlsten, Samuel and Steiner [22], and (2) obtain new metric and topological results on survivor sets of intermediate β -transformations with a hole at zero, extending the work of Kalle, Kong, Langeveld and Li [18]. 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 β -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 [15], Tseng [35] and Färm, Persson and Schmeling [11].

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.

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 Sierpinski 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.

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 β -shifts, namely transformations of the form T β , α : x ↦ β x + α mod 1 acting on [ − α / ( β − 1 ) , ( 1 − α ) / ( β − 1 ) ] , where ( β , α ) ∈ Δ is fixed and where Δ ≔ { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 ≤ α ≤ 2 − β } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of ( β , α ) such that T β , α has the subshift of finite type property is dense in the parameter space Δ . Here, they proposed the following question. Given a fixed β ∈ ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of α in the fiber { β } × ( 0 , 2 − β ) , so that T β , α 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 sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when α = 0 to the case when α ∈ ( 0 , 2 − β ) . That is, we examine the structure of the set of eventually periodic points of T β , α when β is a Pisot number and when β is the n-th root of a Pisot number.

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.

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.

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.

In 2002 Freiberg and Zahle 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.

Given an $\unicode[STIX]{x1D6FC}>1$ and a $\unicode[STIX]{x1D703}$ with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope $\unicode[STIX]{x1D703}$ with respect to (i) an $\unicode[STIX]{x1D6FC}$ -Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of $\unicode[STIX]{x1D703}$ , and (iii) complexity notions which we call $\unicode[STIX]{x1D6FC}$ -repetitiveness, $\unicode[STIX]{x1D6FC}$ -repulsiveness and $\unicode[STIX]{x1D6FC}$ -finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is $2/(\unicode[STIX]{x1D6FC}+1)$ .

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) is an element of R-2 : beta is an element of (1, 2) and 0 <= alpha <= 2 - beta}. This generalises the classical result of Parry from 1960 for greedy beta-shifts.

Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kreĭn, given an atomless Borel probability measure η supported on a compact subset of ℝ, Freiberg and Zähle introduced a measure-geometric approach to define a first order differential operator ∇_η and a second order differential operator Δ_η, with respect to η. We generalise this approach to measures of the form η = ν + δ, where ν is continuous and δ is finitely supported. We determine analytic properties of ∇_η and Δ_η and show that Δ_η 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 Δ_η. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.

A bstract . We determine the structure of the set of intermediate β -shifts of finite type. Specifically, we show that this set is dense in the parameter space ∆ (cid:66) { ( β,α ) ∈ R 2 : β ∈ (1 , 2) and 0 ≤ α ≤ 2 − β } . This generalises the classical result of Parry from 1960 for greedy β -shifts.

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 (\(\alpha \ge 1\)), 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.

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.