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Complex dynamics is the study of iteration of functions which map the complex plane onto itself. In general, their dynamics are quite complicated and hard to explain but for some simple classes of functions many interesting results can be proved. For example, one often studies the class of rational functions (i.e. quotients of polynomials) or, even more specifically, polynomials. Each such function f partitions the extended complex plane C into two regions, one where iteration of the function is chaotic and one where it is not. The nonchaotic region, called the Fatou Set, is the set of all points z such that, under iteration by f, the point z and all its neighbors do approximately the same thing. The remainder of the complex plane is called the Julia set and consists of those points which do not behave like all closely neighboring points. The Julia set of a polynomial typically has a complicated, self similar structure. Many questions can be asked about this structure. The one that we seek to investigate is the notion of the dimension of the Julia set. While the dimension of a line segment, disc, or cube is familiar, there are sets for which no integer dimension seems reasonable. The notion of Hausdorff dimension gives a reasonable way of assigning appropriate non-integer dimensions to such sets. Our goal is to investigate the behavior of the Hausdorff dimension of the Julia sets of a certain simple class of polynomials, namely fd,c(z) = zd + c. In particular, we seek to determine for what values of c and d the Hausdorff dimension of the Julia set varies continuously with c. Roughly speaking, given a fixed integer d > 1 and some complex c, do nearby values of c have Julia sets with Hausdorff dimension relatively close to each other? We find that for most values of c, the Hausdorff dimension of the Julia set does indeed vary continuously with c. However, we shall also construct an infinite set of discontinuities for each d. Our results are summarized in Theorem 10, Chapter 2. In Chapter 1 we state and briefly explain the terminology and definitions we use for the remainder of the paper. In Chapter 2 we will state the main theorems we prove later and deduce from them the desired continuity properties. In Chapters 3 we prove the major results of this paper.

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Several properties of Hausdorff continua are considered in this paper. However, the major emphasis is on developing the properties of the hyperspaces 2x and C(X) of a Hausdorff continuum X. Preliminary definitions and notation are introduced in Chapter I. Chapters II and III deal with the topological structure of the hyperspaces and the concept of topological convergence. Properties of 2x and C(X) are investigated in Chapter IV, while Chapters V and VI are devoted to the Hausdorff continuum X. Chapter VII consists of theorems pertaining to Whitney maps and order arcs in 2x. Examples of C(X) are provided in Chapter VIII. Inverse sequences of Hausdorff continua and of their hyperspaces are considered in Chapter IX.

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by Lee Tin Wah. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 68-69). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Preliminaries --- p.8 / Chapter 2.1 --- Box dimension and Hausdorff dimension --- p.8 / Chapter 2.2 --- Basic properties of dimensions --- p.9 / Chapter 2.3 --- Calculating dimensions --- p.11 / Chapter 3 --- Dimension of graph of the Weierstrass function --- p.14 / Chapter 3.1 --- Calculating dimensions of a graph --- p.14 / Chapter 3.2 --- Weierstrass function --- p.16 / Chapter 3.3 --- An almost everywhere argument --- p.23 / Chapter 3.4 --- Tagaki function --- p.26 / Chapter 4 --- Self-affine mappings --- p.30 / Chapter 4.1 --- Box dimension of self-affine curves --- p.30 / Chapter 4.2 --- Differentability of self-affine curves --- p.35 / Chapter 4.3 --- Tagaki function --- p.42 / Chapter 4.4 --- Hausdorff dimension of self-affine sets --- p.43 / Chapter 5 --- Recurrent set and Weierstrass-like functions --- p.56 / Chapter 5.1 --- Recurrent curves --- p.56 / Chapter 5.2 --- Recurrent sets --- p.62 / Chapter 5.3 --- Weierstrass-like functions from recurrent sets --- p.64 / Bibliography

Consider a family of cubic parabolic polynomials given by for non-zero complex parameters such that for each the polynomial is a parabolic polynomial, that is, the polynomial has a parabolic fixed point and the Julia set of , denoted by , does not contain any critical points of . We also assumed that for each , one finite critical point of the polynomial escapes to the super-attracting fixed point infinity. So, the Julia sets are disconnected. The concern about the family is that the members of this family are generally not even bi-Lipschitz conjugate on their Julia sets. We have proved that the parameter set is open and contains a deleted neighborhood of the origin 0. Our main result is that the Hausdorff dimension function defined by is real analytic. To prove this we have constructed a holomorphic family of holomorphic parabolic graph directed Markov systems whose limit sets coincide with the Julia sets of polynomials up to a countable set, and hence have the same Hausdorff dimension. Then we associate to this holomorphic family of holomorphic parabolic graph directed Markov systems an analytic family, call it , of conformal graph directed Markov systems with infinite number of edges in order to reduce the problem of real analyticity of Hausdorff dimension for the given family of polynomials to prove the corresponding statement for the family .

This thesis makes novel contributions to a problem of practical and theoretical importance, namely how to determine explicitly computable upper bounds for the Hausdorff distance of the spectra of two compact operators on a Hilbert space in terms of the distance of the two operators in operator norm. It turns out that the answer depends crucially on the speed of decay of the sequence of singular values of the two operators. To this end, 'compactness classes', that is, collections of operators the singular values of which decay at a certain speed, are introduced and their functional analytic properties studied in some detail. The main result of the thesis is an explicit formula for the Hausdorff distance of the spectra of two operators belonging to the same compactness class. Along the way, upper bounds for the resolvents of operators belonging to a particular compactness class are established, as well as novel bounds for determinants of trace class operators.