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|Title: ||Real and Complex Dynamics of Unicritical Maps|
|Authors: ||Clark, Trevor Collin|
|Advisor: ||Lyubich, Mikhail|
|Keywords: ||holomorphic dynamics|
|Issue Date: ||6-Aug-2010|
|Abstract: ||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.|
|Appears in Collections:||Doctoral|
Department of Mathematics - Doctoral theses
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