This dissertation is in the area of complex dynamics, more specifically focused on the iteration of rational functions. Given a well-chosen family of rational functions, parameterized by a complex parameter, we are especially interested in regularity properties of the Hausdorff dimension of Julia sets of these polynomials considered as a function of the parameters. In this dissertation I deal with a family of polynomials of degree at least 3 depending in a holomorphic way on a parameter, focusing on the point where the dynamics and topology of the polynomials drastically change. In such a context proving continuity is quite challenging while real analyticity will most likely break. Our approach will, on the one hand, build on the existing methods of proving continuity of Hausdorff dimension, primarily based on proving continuity, in the weak* topology of measures on the Riemann sphere, of canonical conformal measures, but will also require methods which, up to my best knowledge, have not been implemented anywhere yet. Our main result gives a surprising example where the Hausdorff dimension of the Julia set is continuous in the parameter, but where the Julia set itself is not.

For the family of complex rational maps F_λ(z)=z^n+λ/z^d, where λ is a complex parameter and n, d ≥ 2 are integers, many small copies of the well-known Mandelbrot set are visible in the parameter plane. An infinite number of these are located around the boundary of the connectedness locus and are accessible by parameter rays from the Cantor set locus. Maps taken from main cardioid of these accessbile Mandelbrot sets have attracting periodic cycles. A method for constructing models of the Julia sets corresponding to such maps is described. These models are then used to explore the existence of dynamical conjugacies between maps taken from distinct accessible Mandelbrot sets in the upper halfplane.

For parametrized families of dynamical systems, two major goals are classifying the systems up to topological conjugacy, and understanding the structure of the bifurcation locus. The family Fλ = z^n + λ/z^d gives a 1-parameter, n+d degree family of rational maps of the Riemann sphere, which arise as singular perturbations of the polynomial z^n. This work presents several results related to these goals for the family Fλ, particularly regarding a structure of "necklaces" in the λ parameter plane. This structure consists of infinitely many simple closed curves which surround the origin, and which contain postcritically finite parameters of two types: superstable parameters and escape time Sierpinski parameters. First, we derive a dynamical invariant to distinguish the conjugacy classes among the superstable parameters on a given necklace, and to count the number of conjugacy classes. Second, we prove the existence of a deeper fractal system of "subnecklaces," wherein the escape time Sierpinski parameters on the previously known necklaces are themselves surrounded by infinitely many necklaces.

We identify three structures that lie in the parameter plane of the rational map F(z) = zⁿ + λ / zᵈ, for which z is a complex number, λ a complex parameter, n ≥ 4 is even, and d ≥ 3 is odd. There exists a Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the end of the arc. There exists as well a qualitatively different Sierpindelbrot arc, an infinite sequence of pairs of Mandelbrot sets and Sierpinski holes, that limits to the parameter at the center of the arc. Furthermore, there exist infinitely many arcs of each type. A parameter can travel along a continuous path from the Cantor set locus, along infinitely many arcs of the first type in a successively smaller region of the parameter plane, while passing through an arc of the second type, to the parameter at the center of the latter arc. This infinite sequence of Sierpindelbrot arcs is a Sierpinski Mandelbrot spiral.

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It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational map of degree at least two and if ϕ:ℂ→ℝ is Hölder continuous and satisfies the “thermodynamic expanding” condition P(T,ϕ) > sup(ϕ), then there exists exactly one equilibrium state μ for T and ϕ, and furthermore (ℂ,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on ℂ, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz. Specifically, if (T,Ω,P,θ) is a holomorphic random dynamical system on ℂ and ϕ:Ω→ ℋα(ℂ) is a Hölder continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of (X,P,ϕ) over (Ω,Ρ,θ).

The Brjuno numbers play a fundamental role in the study of the 1-dimensional Complex Dynamics Theory. In this work we examine the proof of the Brjuno theorem by using elements of Number Theory. We also examine the topological version of the proof given by J. Yoccoz and his renormalization principle. If α ∈ ℝ\ℚ, we also describe how the existence of a Siegel disk at the origin for the polynomial P(𝑧) = exp(2πiα)·(𝑧 − 𝑧²) implies the linearization of any germ of the form 𝑓(𝑧) = exp(2πiα)·𝑧 + 𝑂(𝑧²). / Master of Science

Both complex dynamics and the theory of reproducing kernel Hilbert spaces have found widespread application over the last few decades. Although complex dynamics started over a century ago, the gravity of it's importance was only recently realized due to B.B. Mandelbrot's work in the 1980's. B.B. Mandelbrot demonstrated to the world that fractals, which are chaotic patterns containing a high degree of self-similarity, often times serve as better models to nature than conventional smooth models. The theory of reproducing kernel Hilbert spaces also having started over a century ago, didn't pick up until N. Aronszajn's classic was written in 1950. Since then, the theory has found widespread application to fields including machine learning, quantum mechanics, and harmonic analysis. In the paper, Infinite Product Representations of Kernel Functions and Iterated Function Systems, the authors, D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martiziano, show how a kernel function can be constructed on an attracting set of an iterated function system. Furthermore, they show that when certain conditions are met, one can construct an orthonormal basis of the associated Hilbert space via certain pull-back and multiplier operators. In this thesis we take for our iterated function system, the family of iterates of a given rational map. Thus we investigate for which rational maps their kernel construction holds as well as their orthornormal basis construction. We are able to show that the kernel construction applies to any rational map conjugate to a polynomial with an attracting fixed point at 0. Within such rational maps, we are able to find a family of polynomials for which the orthonormal basis construction holds. It is then natural to ask how the orthonormal basis changes as the polynomial within a given family varies. We are able to determine for certain families of polynomials, that the dynamics of the corresponding orthonormal basis is well behaved. Finally, we conclude with some possible avenues of future investigation.

Indiana University-Purdue University Indianapolis (IUPUI) / Given a parametrized family of Iterated Function System (IFS) we give sufficient conditions for a parameter on the boundary of the connectedness locus, M, to be accessible from the complement of M. Moreover, we provide a few examples of such parameters and describe how they are connected to Misiurewicz parameter in the Mandelbrot set, i.e. the connectedness locus of the quadratic family z^2+c.