40 research outputs found
Convex Dynamics and Applications
This paper proves a theorem about bounding orbits of a time dependent
dynamical system. The maps that are involved are examples in convex dynamics,
by which we mean the dynamics of piecewise isometries where the pieces are
convex. The theorem came to the attention of the authors in connection with the
problem of digital halftoning. \textit{Digital halftoning} is a family of
printing technologies for getting full color images from only a few different
colors deposited at dots all of the same size. The simplest version consist in
obtaining grey scale images from only black and white dots. A corollary of the
theorem is that for \textit{error diffusion}, one of the methods of digital
halftoning, averages of colors of the printed dots converge to averages of the
colors taken from the same dots of the actual images. Digital printing is a
special case of a much wider class of scheduling problems to which the theorem
applies. Convex dynamics has roots in classical areas of mathematics such as
symbolic dynamics, Diophantine approximation, and the theory of uniform
distributions.Comment: LaTex with 9 PostScript figure
Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors
We consider two dimensional maps preserving a foliation which is uniformly
contracting and a one dimensional associated quotient map having exponential
convergence to equilibrium (iterates of Lebesgue measure converge exponentially
fast to physical measure). We prove that these maps have exponential decay of
correlations over a large class of observables. We use this result to deduce
exponential decay of correlations for the Poincare maps of a large class of
singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos
corrected; improvements on the statements and comments suggested by a
referee. Keywords: singular flows, singular-hyperbolic attractor, exponential
decay of correlations, exact dimensionality, logarithm la
Entropic Fluctuations in Statistical Mechanics I. Classical Dynamical Systems
Within the abstract framework of dynamical system theory we describe a
general approach to the Transient (or Evans-Searles) and Steady State (or
Gallavotti-Cohen) Fluctuation Theorems of non-equilibrium statistical
mechanics. Our main objective is to display the minimal, model independent
mathematical structure at work behind fluctuation theorems. Besides its
conceptual simplicity, another advantage of our approach is its natural
extension to quantum statistical mechanics which will be presented in a
companion paper. We shall discuss several examples including thermostated
systems, open Hamiltonian systems, chaotic homeomorphisms of compact metric
spaces and Anosov diffeomorphisms.Comment: 72 pages, revised version 12/10/2010, to be published in Nonlinearit
The Conley Conjecture and Beyond
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.Comment: 34 pages, 1 figur
Wilkinson's work was partly funded by NSF Grant #DMS-0100314.
A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of dierentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself. Dedicated to David Ruelle on his 65th birthday. October 15, 2002 Shub's work was partly funded by NSF Grant #DMS-9988809. Wilkinson's work was partly funded by NSF Grant #DMS-0100314. 1