In this thesis we study the moduli of holomorphic vector bundles over a non-compact complex space X, which will mainly be of dimension 2 or 3 and which contains a distinguished rational curve ℓ ⊂ X. We will consider the situation in which X is the total space of a holomorphic vector bundle on CP1 and ℓ is the zero section. While the treatment of the problem in this full generality requires the study of complex analytic spaces, it soon turns out that a large part of it reduces to algebraic geometry. In particular, we prove that in certain cases holomorphic vector bundles on X are algebraic. A key ingredient in the description of themoduli are numerical invariants that we associate to each holomorphic vector bundle. Moreover, these invariants provide a local version of the second Chern class. We obtain sharp bounds and existence results for these numbers. Furthermore, we find a new stability condition which is expressed in terms of these numbers and show that the space of stable bundles forms a smooth, quasi-projective variety.

In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincare exponent for the Fibonacci map is less than two, which implies that the Hausdor ff dimension of its Julia set is less than two.

In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps. In the first part, we construct a lamination of the space of unimodal maps whose critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the parameter space for one-parameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space. This result implies that the statistical description of typical unimodal maps obtained in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in particular, almost every map in such a family is either regular or stochastic. In the second part, we prove the Poincare exponent for the Fibonacci map is less than two, which implies that the Hausdor ff dimension of its Julia set is less than two.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 68-70).

Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 100-102).

published_or_final_version / abstract / toc / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (Ph. D.)--Missouri University of Science and Technology, 2009. / Vita. The entire thesis text is included in file. Title from title screen of thesis/dissertation PDF file (viewed April 29, 2009) Includes bibliographical references (p. 142-144).

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 68-70).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf 32).