Disintegration of invariant measures for hyperbolic skew products

We study hyperbolic skew products and the disintegration of the SRB measure into measures supported on local stable manifolds. Such a disintegration gives a method for passing from an observable v on the skew product to an observable \(\overline v \) on the system quotiented along stable manifolds. Under mild assumptions on the system we prove that the disintegration preserves the smoothness of v, first in the case where v is Hölder and second in the case where v is \({\mathcal{C}^1}\).     

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata’s Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of uniform dual ergodicity for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides second (and higher) order asymptotics. In the process, we derive a higher order Tauberian theorem for scalar power series which, to our knowledge, has not previously been covered.     

X-ray structure of 1,3-adamantane dicarbonamide

The X-ray crystal structure of 1,3-adamantane-dicarbonamide has been determined. Colourless, cube shaped crystals of C₁₂H₁₈N₂O₂ crystallize in the monoclinic space group C2/c (#15) with cell dimensionsa=7·934(3),b=15·543(3),c=9·404(2) Å, =93.63°;V=1157.4(5) ų andZ=4.1074 unique reflections withI _net >2.00(I) on refinement afforded values ofR=0.039 andR _w =0.040.     

Bifurcation from Discrete Rotating Waves

Discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries. We present a systematic approach to the study of local bifurcation from discrete rotating waves. The approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range. Our results are valid for dynamical systems with finite symmetry group, and more generally, for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group.     

Steady-state bifurcation with Euclidean symmetry

We consider systems of partial differential equations equivariant under the Euclidean group and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when and and for reaction-diffusion equations with general , reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.)

In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of . The representation theory of is driven by the irreducible representations of . For , this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.)

When , there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of . There are infinitely many possibilities for each .

     

X-ray structure of tetramethyl thiophenetetracarboxylate

The X-ray crystal structure of tetramethyl thiophenetetracarboxylate has been determined. Colourless, plates of C₁₂H₁₂O₈S crystallise in the orthorhombic space group Pbca with cell dimensionsa=18.321(7),b=17.98(1),c=8.675(6)Å;V=2858(5)ų andz=8.1568 unique reflections withI _net>2(I) on refinement afforded values ofR=0.040 andR _w=0.034.     

Stable Transitivity of Certain Noncompact Extensions of Hyperbolic Systems

Let be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finite-dimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive -extensions. More generally, we find stably transitive -extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive.For groups of the form where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalizes a result of Niţică and Pollicott for -extensions.Communicated by Viviane Baladisubmitted 24/05/04, accepted 11/10/04