1,141 research outputs found
Linear fractional transformations mod one and ergodic theory
Imperial Users onl
Universal geometric approach to uncertainty, entropy and information
It is shown that for any ensemble, whether classical or quantum, continuous
or discrete, there is only one measure of the "volume" of the ensemble that is
compatible with several basic geometric postulates. This volume measure is thus
a preferred and universal choice for characterising the inherent spread,
dispersion, localisation, etc, of the ensemble. Remarkably, this unique
"ensemble volume" is a simple function of the ensemble entropy, and hence
provides a new geometric characterisation of the latter quantity. Applications
include unified, volume-based derivations of the Holevo and Shannon bounds in
quantum and classical information theory; a precise geometric interpretation of
thermodynamic entropy for equilibrium ensembles; a geometric derivation of
semi-classical uncertainty relations; a new means for defining classical and
quantum localization for arbitrary evolution processes; a geometric
interpretation of relative entropy; and a new proposed definition for the
spot-size of an optical beam. Advantages of the ensemble volume over other
measures of localization (root-mean-square deviation, Renyi entropies, and
inverse participation ratio) are discussed.Comment: Latex, 38 pages + 2 figures; p(\alpha)->1/|T| in Eq. (72) [Eq. (A10)
of published version
Dynamic isoperimetry and the geometry of Lagrangian coherent structures
The study of transport and mixing processes in dynamical systems is
particularly important for the analysis of mathematical models of physical
systems. We propose a novel, direct geometric method to identify subsets of
phase space that remain strongly coherent over a finite time duration. This new
method is based on a dynamic extension of classical (static) isoperimetric
problems; the latter are concerned with identifying submanifolds with the
smallest boundary size relative to their volume.
The present work introduces \emph{dynamic} isoperimetric problems; the study
of sets with small boundary size relative to volume \emph{as they are evolved
by a general dynamical system}. We formulate and prove dynamic versions of the
fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming
theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian
operator and describe a computational method to identify coherent sets based on
eigenfunctions of the dynamic Laplacian.
Our results include formal mathematical statements concerning geometric
properties of finite-time coherent sets, whose boundaries can be regarded as
Lagrangian coherent structures. The computational advantages of our new
approach are a well-separated spectrum for the dynamic Laplacian, and
flexibility in appropriate numerical approximation methods. Finally, we
demonstrate that the dynamic Laplacian operator can be realised as a
zero-diffusion limit of a newly advanced probabilistic transfer operator method
(Froyland, 2013) for finding coherent sets, which is based on small diffusion.
Thus, the present approach sits naturally alongside the probabilistic approach
(Froyland, 2013), and adds a formal geometric interpretation
Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
In this paper we develop new techniques for revealing geometrical structures
in phase space that are valid for aperiodically time dependent dynamical
systems, which we refer to as Lagrangian descriptors. These quantities are
based on the integration, for a finite time, along trajectories of an intrinsic
bounded, positive geometrical and/or physical property of the trajectory
itself. We discuss a general methodology for constructing Lagrangian
descriptors, and we discuss a "heuristic argument" that explains why this
method is successful for revealing geometrical structures in the phase space of
a dynamical system. We support this argument by explicit calculations on a
benchmark problem having a hyperbolic fixed point with stable and unstable
manifolds that are known analytically. Several other benchmark examples are
considered that allow us the assess the performance of Lagrangian descriptors
in revealing invariant tori and regions of shear. Throughout the paper
"side-by-side" comparisons of the performance of Lagrangian descriptors with
both finite time Lyapunov exponents (FTLEs) and finite time averages of certain
components of the vector field ("time averages") are carried out and discussed.
In all cases Lagrangian descriptors are shown to be both more accurate and
computationally efficient than these methods. We also perform computations for
an explicitly three dimensional, aperiodically time-dependent vector field and
an aperiodically time dependent vector field defined as a data set. Comparisons
with FTLEs and time averages for these examples are also carried out, with
similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure
On the arithmetic of crossratios and generalised Mertens' formulas
We develop the relation between hyperbolic geometry and arithmetic
equidistribution problems that arises from the action of arithmetic groups on
real hyperbolic spaces, especially in dimension up to 5. We prove
generalisations of Mertens' formula for quadratic imaginary number fields and
definite quaternion algebras over the rational numbers, counting results of
quadratic irrationals with respect to two different natural complexities, and
counting results of representations of (algebraic) integers by binary
quadratic, Hermitian and Hamiltonian forms with error bounds. For each such
statement, we prove an equidistribution result of the corresponding
arithmetically defined points. Furthermore, we study the asymptotic properties
of crossratios of such points, and expand Pollicott's recent results on the
Schottky-Klein prime functions.Comment: 44 page
Smoothness, asymptotic smoothness and the Blum-Hanson property
We isolate various su cient conditions for a Banach space X to have the so-called Blum-Hanson property. In particular, we show that X has the Blum-Hanson property if either the modulus of asymptotic smoothness of X has an extremal behaviour at in nity, or if X is uniformly G^ateaux smooth and embeds isometrically into a Banach space with a 1-unconditional nite-dimensional decomposition
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