Discrete-time topological dynamics, complex Hadamard matrices, and oblique-incidence ion bombardment - Motta, Francis Charles, author; Shipman, Patrick D., advisor; Dangelmayr, Gerhard, committee member; Peterson, Chris, committee member; Bradley, Mark, committee member (Colorado State University. Libraries, 2014)
The topics covered in this dissertation are not unified under a single mathematical discipline. However, the questions posed and the partial solutions to problems of interest were heavily influenced by ideas from dynamical systems, mathematical experimentation, and simulation. Thus, the chapters in this document are unified by a common flavor which bridges several mathematical and scientific disciplines. The first chapter introduces a new notion of orbit density applicable to discrete-time dynamical systems on a topological phase space, called the linear limit density of an orbit. For a fixed discrete-time dynamical system, Φ(χ) : M → M defined on a bounded metric space, we introduce a function E : {γχ : χ ∈ Mg} → R∪{∞} on the orbits of Φ, γχ ≐ {Φt(χ) : t ∈ N}, and interpret E(γχ) as a measure of the orbit's approach to density; the so-called linear limit density (LLD) of an orbit. We first study the family of dynamical systems Rθ : [0; 1) → [0; 1)(θ ∈ (0; 1)) defined by Rθ(χ) = (χ + θ) mod 1. Utilizing a formula derived from the Three-Distance theorem, we compute the exact value of E({RtΦ(χ) : t ∈ N}, χ ∈ [0; 1)), where Φ = √5 – 1) /2. We further compute E({Rtθ(χ) : t ∈ N}; χ ∈ [0, 1)) for a class of irrational rotation angles θ = [j, j,…] with period-1 continued fraction expansions and discuss how this measure distinguishes the topologically transitive behavior of different choices of θ. We then expand our focus to a much broader class of orientation-preserving homeomorphisms of the circle and extend a result of R. Graham and J.H. van Lint about optimal irrational rotations. Finally, we consider the LLD of orbits of the Bernoulli shift map acting on sequences defined over a finite alphabet and prove bounds for a class of sequences built by recursive extension of de Bruijn sequences. To compute approximations of E(γχ) for orbits of the Bernoulli shift map, we develop an efficient algorithm which determines a point in the set of all words of a fixed length over a finite alphabet whose distance to a distinguished subset is maximal. Chapter two represents a departure from a dynamical systems problem by instead exploring the structure of the space of complex Hadamard matrices and mutually unbiased bases (MUBs) of complex Hilbert space. Although the problem is not intrinsically dynamical, our mechanisms for experimentation and exploration include an algorithm which can be viewed as a discrete-time dynamical system as well as a gradient system of ordinary differential equations (ODEs) whose fixed points are dephased complex Hadamards. We use our discrete system to produce numerical evidence which supports existing conjectures regarding complex Hadamards and mutually unbiased bases, including that the maximal size of a set of 6 x 6 MUBs is four. By applying center-manifold theory to our gradient system, we introduce a novel method to analyze the structure of Hadamards near a fixed matrix. In addition to formalizing this technique, we apply it to prove that a particular 9 x 9 Hadamard does not belong to a continuous family of inequivalent matrices, despite having a positive defect. This is the first known example of this type. The third chapter explores the phenomenon of pattern formation in dynamical systems by considering a model of off-normal incidence ion bombardment (OIIB) of a binary material. We extend the Bradley-Shipman theory of normal-incidence ion bombardment of a binary material by analyzing a system of partial differential equations that models the off-normal incidence ion bombardment of a binary material by coupling surface topography and composition. In this chapter we perform linear and non-linear analysis of the equations modeling the interaction between surface height and composition and derive a system of ODEs which govern the time-evolution of the unstable modes, allowing us to identify parameter ranges which lead to patterns of interest. In particular, we demonstrate that an unusual "dots-on-ripples" topography can emerge for nonzero angles of ion incidence θ. In such a pattern, nanodots arranged in a hexagonal array sit atop a ripple topography. We find that if dots-on-ripples are supplanted by surface ripples as θ or the ion energy are varied, the transition is continuous.
Keywords: topological dynamics; center manifold; de Bruijn; Hadamard; ion bombardment; pattern formation