Linda Keen

Chaos and Fractals: The Mathematics behind the Computer Graphics

Overview: Dynamics of simple maps by R. L. Devaney Nonlinear oscillations and the Smale horseshoe map by P. J. Holmes Fractal basin boundaries and chaotic attractors by K. T. Alligood and J. A. Yorke Julia sets by L. Keen The Mandelbrot set by B. Branner Introduction to fractals by J. Harrison Iterated function systems by M. F. Barnsley.

Dynamics of holomorphic self-maps of ℂ*

In this paper we classify the stable components of holomorphic self-maps of ℂ* which have finitely many singular values. We use this to study a one and a two parameter family of such functions. We examine the dynamic dependence of these functions on the parameters and study the parameter spaces themselves.

Dynamics of the family tan

We study the dynamics of the tangent family z -> lambda tan z for lambda complex and give a complete classification of their stable behavior. We also characterize the the hyperbolic components and give a combinatorial description their deployment in the parameter plane.

On Fundamental Domains and the Teichmüller Modular Group

Publisher Summary This chapter discusses the problem of determining a fundamental domain for M(G) and presents steps to solve it, where M is modulo group of orientation preserving homeomorphisms of S modulo those isotopic to the identity and G is a Fuchsian group. The fundamental domains are explored for the action of M on the Teichmuller spaces of the simple surfaces and for the action of M on the combination or twist parameters. The chapter also presents a proof of the existence of bounds on the lengths of certain classes of geodesies on S.

Discreteness criteria and the hyperbolic geometry of palindromes

We consider non-elementary representations of two generator free groups in PSL(2;C), not necessarily discrete or free, G = hA;Bi. A word in A and B, W (A;B), is a palindrome if it reads the same forwards and backwards. A word in a free group is primitive if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identied with the positive rational numbers. We study the geometry of palindromes and the action of G in H3 whether or not G is discrete. We show that there is a core geodesic L in the convex hull of the limit set of G and use it to prove three results: the rst is that there are well dened maps from the non- negative rationals and from the primitive elements to L; the second is that G is geometrically nite if and only if the axis of every non-parabolic palindromic word in G intersects L in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.

Hyperbolic Geometry from a Local Viewpoint: Contents

Introduction 1. Elementary transformations 2 Hyperbolic metric in the unit disk 3. Holomorphic functions 4. Topology and uniformization 5. Discontinuous groups 6 Fuchsian groups 7. General hyperbolic metric 8. The Kobayashi metric 9. The Caratheodory pseudo metric 10. Contraction properties 11. Applications 12 Applications II 13. Applications III 14. Estimating hyperbolic densities 15. Uniformly perfect domains 16 Appendix: Elliptic functions Bibliography.

Bounded-type Siegel disks of a one-dimensional family of entire functions

Let 0< θ < 1 be an irrational number of bounded type. We prove that for any map in the family (e2π iθ z + αz2)ez , α ∈ C, the boundary of the Siegel disk, with fixed point at the origin, is a quasi-circle passing through one or both of the critical points.

A modular group and Riemann surfaces of genus 2

Let T(S) be the Teichmiiller space of Compact Riemann surfaces of genus 2 and let M(S) be the corresponding Teichmiiller modular group. In this paper, we determine a fundamental region R(S) for M(S) acting on T(S). R(S) is constructed as a subspace of T(S), as it is described in [8-10]. We make use of and extend the results in [12] to fully describe the action of M(S) on the parameters determining T(S). Our construction also depends strongly on a theorem of Bers [3]. In Section I, we recall some basic definitions, and determine our notation. In Section II, we describe the moduli space for a torus with a hole, the action of the corresponding modular group on this space, and determine a fundamental domain for this modular group. In Section III we describe the space T(S) and in Section IV we discuss the modular group and study its action on T(S). In Section V we state the theorem of Bers in preparation for the actual construction in Section V1. Finally, in Section VII we use the constructed fundamental domain to prove a conjecture of Bets in [2] for the case of Riemann surfaces of genus 2.

Random holomorphic iterations and degenerate subdomains of the unit disk

Given a random sequence of holomorphic maps f 1 , f 2 , f 3 ,... of the unit disk A to a subdomain X, we consider the compositions F n = fl f 2 ... f n-1 0 f n . The sequence {F n } is called the iterated function system coming from the sequence f 1 , f 2 , f 3 .... We prove that a sufficient condition on the domain X for all limit functions of any {F n } to be constant is also necessary. We prove that the condition is a quasiconformal invariant. Finally, we address the question of uniqueness of limit functions.

The set of maps\(F_{a,b} :x \mapsto x + a + \tfrac{b}{{2\pi }}\) sin(2πx) with any given rotation interval is contractiblewith any given rotation interval is contractible

AbstractConsider the two-parameter family of real analytic maps