A NON-EXHAUSTIVE TOPICS LIST, MATH 3062/6116, SEMESTER 1 2021 This list is not necessarily complete. I will probably add to over the weekend… (1) Transformations and spaces. Definitions, some basic properties. (2) Affine transformations Fitting affine transformations, i.e. the fundamental theorem of affine ge- ometry. (3) Similitudes Fitting similitudes (4) Metric spaces Definition of metric space. The triangle inequality and how to use it. (5) Basic topology Concepts like open balls. Definitions of convergence, continuity, complete- ness, compactness as defined in metric spaces. Familiarity with Heine-Borel / Bolzano-Weierstrass principles for closed and bounded sets in Rn. (6) Contraction mappings Definition of contraction mappings. (7) The Banach contraction mapping theorem / fixed point theorem Definition of fixed points. (HPO/6116) Familiarity with the concepts be- hind the proof (8) The Hausdorff metric Definition of the Hausdorff metric and the equivalent definition. Set dila- tions. (9) Iterated function systems (IFSs) Definition of the Hutchinson operator. (10) Hutchinson’s theorem The existence of the unique attractor A of a contractive IFS. a.k.a. the contractivity of a contractive IFS in Hausdorff metric + Banach contraction mapping theorem. (11) The chaos game with IFSs (12) The collage theorem And, how to fit an IFS to a picture This is roughly “half way” here: there will be just slightly more of an emphasis on the material after this point (13) Codespace and the codespace metric Definition including infinite and finite codes. (14) Addressing on attractors and the pi map 1 2 A NON-EXHAUSTIVE TOPICS LIST, MATH 3062/6116, SEMESTER 1 2021 Ensure you are familiar with the broad concepts behind finite addressing and infinite addressing. (15) Fractal dimension The box counting dimension, and an understanding that we can count boxes or balls (or any other regular shape). (16) (HPO / 6116) Hausdorff dimension (17) Dynamical systems Definitions. Know how to draw cobweb diagrams. (18) Chaotic dynamical systems Particularly Devaney’s definition of chaos (19) The logistic map The logistic map on R, f(x) = μx(1 x). The attractive periodic points of the logistic map and the cascade of bifurcations. The conjugacy of the logistic map on C, fμ(z) = μz(1 z) and the quadratic gλ(z) = z2 λ. (Warning! The labels μ and λ have been interchanged a few times in the notes, make sure you know which is whihch at all times!) (20) The shift map on codespace The theorem that the shift map is chaotic on codespace. (21) The shift map on the attractor of an IFS How does the shift map correspond to the movement of a point on an attractor (22) Julia and Fatou sets Their definitions. (HPO/6116) Familiarity with existence, compactness, and invariance proofs. (23) Newton’s method Definition of Newton’s method. Seeing Newton’s method as a dynamical system. Roots of polynomials as attractive fixed points of Newton’s method. The existence of “Julia” sets in the boundaries between the basins of attrac- tion. (24) The Mandlebrot set The two equivalent definitions of the Mandlebrot set. (HPO / 6116) Some understanding of the bifurcations of the logistic map in relation to the Man- dlebrot set. (25) Fractal interpolation Familiarity with the method and the goals of fractal interpolation. Calcu- lating the dimension. (26) Brownian motion Familiarity with the concept of Brownian motion and how it might be constructed.
