A Note on Operator Semigroups Associated to Chaotic Flows

The transfer operator associated to a flow (continuous time dynamical system) is a one-parameter operator semigroup. We consider the operator-valued Laplace transform of this one-parameter semigroup. Estimates on the Laplace transform have be used in various settings in order to show the rate at which the flow mixes. Here we consider the case of exponential mixing or rapid mixing (super polynomial). We develop the operator theory framework amenable to this setting and show that the same estimates may be used to produce results, in terms of the operators, which go beyond the results for the rate of mixing. Such results are useful for obtaining other statistical properties of the dynamical system

An Alternative Approach to Generalised BV and the Application to Expanding Interval Maps

We introduce a family of Banach spaces of measures, each containing the set of measures with density of bounded variation. These spaces are suitable for the study of weighted transfer operators of piecewise-smooth maps of the interval where the weighting used in the transfer operator is not better than piecewise H\"older continuous and the partition on which the map is continuous may possess a countable number of elements. For such weighted transfer operators we give upper bounds for both the spectral radius and for the essential spectral radius

Parabolic Flows Renormalized by Partially Hyperbolic Maps

We consider parabolic flows on 3-dimensional manifolds which are renormalized by circle extensions of Anosov diffeormorphisms. This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalized by partially hyperbolic automorphisms. The transfer operators associated to the renormalization maps, acting on anisotropic Sobolev spaces, are known to have good spectral properties (this relies on ideas which have some resemblance to representation theory but also apply to non-algebraic systems). The spectral information is used to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow.Comment: Comments welcom

Open sets of Axiom A flows with Exponentially Mixing Attractors (with Erratum)

For any dimension $d\geq 3$ we construct $C^{1}$-open subsets of the space of $C^{3}$ vector fields such that the flow associated to each vector field is Axiom A and exhibits a non-trivial attractor which mixes exponentially with respect to the unique SRB measure.Comment: 15 pages, no figures; content as published in journal; includes text of the Erratum as an appendi

Laser guide stars and turbulence profiling for extremely large telescopes

The next generation of ground based telescopes, the so-called extremely large telescopes, will offer a significant leap in sensitivity and resolution compared to current telescopes. They also present a range of technical challenges. This thesis presents work on two important problems in implementing laser guide star adaptive optics on extremely large telescopes: focal anisoplanatism and turbulence profiling. The SPLASH (Sky-Projected Laser Array Shack-Hartmann) laser guide star wavefront sensing technique is described. The technique is shown to offer reduced focal anisoplanatism compared to a conventional laser guide star for large telescopes. The technique may also offer advantages for larger apertures, including extremely large telescopes, but simulations were limited to 8 metre apertures by currently available computing capabilities. A calibration method is presented for the SLODAR (Slope Detection and Ranging) turbulence profiling technique, along with an analysis of the effects of scintillation on SLODAR when the technique is applied on a small (~ 40 cm diameter)' telescope. A new variation on the SLODAR technique, SLOTDAR (Slope Detection and Ranging through a slot), is introduced, in which the spatial sampling can be optimised based on the brightness of the available reference stars

Anisotropic spaces and nilmanifold automorphisms

We introduce anisotropic Banach spaces on Heisenberg nilmanifolds and study the resonance spectrum associated to partially hyperbolic automorphisms. In this work we describe an alternative proof of the spectrum which takes advantage of geometric-style anisotropic norms.Comment: Comments welcom