T-Space Collection:http://hdl.handle.net/1807/253362014-03-24T06:47:22Z2014-03-24T06:47:22ZReal and Complex Dynamics of Unicritical MapsClark, Trevor Collinhttp://hdl.handle.net/1807/247192010-11-03T20:56:44Z2010-08-06T17:39:05ZTitle: Real and Complex Dynamics of Unicritical Maps
Authors: Clark, Trevor Collin
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.2010-08-06T17:39:05ZTowards a Bezout-type Theory of Affine VarietiesMondal, Pinakihttp://hdl.handle.net/1807/243712010-12-10T21:43:45Z2010-04-21T18:48:56ZTitle: Towards a Bezout-type Theory of Affine Varieties
Authors: Mondal, Pinaki
Abstract: We study projective completions of affine algebraic varieties (defined over an algebraically closed field K) which are given by filtrations, or equivalently, integer valued `degree like functions' on their rings of regular functions. For a polynomial map P := (P_1, ..., P_n): X -> K^n of affine varieties with generically finite fibers, we prove that there are completions of the source such that the intersection of completions of the hypersurfaces {P_j = a_j} for generic (a_1, ..., a_n) in K^n coincides with the respective fiber (in short, the completions `do not add points at infinity' for P). Moreover, we show that there are `finite type' completions with the latter property, i.e. determined by the maximum of a finite number of `semidegrees', by which we mean degree like functions that send products into sums. We characterize the latter type completions as the ones for which ideal I of the `hypersurface at infinity' is radical. Moreover, we establish a one-to-one correspondence between the collection of minimal associated primes of I and the unique minimal collection of semidegrees needed to define the corresponding degree like function. We also prove an `affine Bezout type' theorem for polynomial maps P with finite fibers that admit semidegrees corresponding to completions that do not add points at infinity for P. For a wide class of semidegrees of a `constructive nature' our Bezout-type bound is explicit and sharp.2010-04-21T18:48:56ZDimension Groups and C*-algebras Associated to Multidimensional Continued FractionsMaloney, Gregoryhttp://hdl.handle.net/1807/243152013-10-11T18:27:29Z2010-04-13T15:34:18ZTitle: Dimension Groups and C*-algebras Associated to Multidimensional Continued Fractions
Authors: Maloney, Gregory
Abstract: Thirty years ago, Effros and Shen classified the simple dimension groups with rank two. Every such group is parametrized by an irrational number, and can be constructed as an inductive limit using that number's continued fraction expansion.
There is a natural generalization of continued fractions to higher dimensions, and this invites the following question: What dimension groups correspond to multidimensional continued fractions? We describe this class of groups and show how some properties of a continued fraction are reflected in the structure of its dimension group.
We also consider a related issue: an Effros-Shen group has been shown to arise in a natural way from the tail equivalence relation on a certain sequence space. We describe a more general class of sequence spaces to which this construction can be applied to obtain other dimension groups, including dimension groups corresponding to multidimensional continued fractions.2010-04-13T15:34:18ZLeaf Conjugacies on the TorusHammerlindl, Andrew Scotthttp://hdl.handle.net/1807/193242010-12-10T20:58:45Z2010-03-10T16:44:26ZTitle: Leaf Conjugacies on the Torus
Authors: Hammerlindl, Andrew Scott
Abstract: If a partially hyperbolic diffeomorphism on a torus of dimension d greater than 3 has
stable and unstable foliations which are quasi-isometric on the universal cover,
and its center direction is one-dimensional, then the diffeomorphism is leaf
conjugate to a linear toral automorphism. In other words, the hyperbolic
structure of the diffeomorphism is exactly that of a linear, and thus simple to
understand, example. In particular, every partially hyperbolic diffeomorphism on
the 3-torus is leaf conjugate to a linear toral automorphism.2010-03-10T16:44:26Z