Lyapunov 1-forms for flows

In this paper we find conditions which guarantee that a given flow $\Phi$ on a compact metric space $X$ admits a Lyapunov one-form $\omega$ lying in a prescribed \v{C}ech cohomology class $\xi\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $\xi$ to the chain recurrent set of $\Phi$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.Comment: 27 pages, 3 figures. This revised version incorporates a few minor improvement

Compactness results in Symplectic Field Theory

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in [M Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307--347] as well as compactness theorems in Floer homology theory, [A Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775--813 and Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988) 513--547], and in contact geometry, [H Hofer, Pseudo-holomorphic curves and Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 307--347 and H Hofer, K Wysocki and E Zehnder, Foliations of the Tight Three Sphere, Annals of Mathematics, 157 (2003) 125--255].Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper25.abs.htm

An inverse function theorem in Fréchet-Spaces

AbstractA technical inverse function theorem of Nash-Moser type is proved for maps between Fréchet spaces allowing smoothing operators. A counterexample shows that the growth requirements on the rightinverse of the linearized map needed are minimal

Almost reducibility for finitely differentiable SL(2,R)-valued quasi-periodic cocycles

Quasi-periodic cocycles with a diophantine frequency and with values in SL(2,R) are shown to be almost reducible as long as they are close enough to a constant, in the topology of k times differentiable functions, with k great enough. Almost reducibility is obtained by analytic approximation after a loss of differentiability which only depends on the frequency and on the constant part. As in the analytic case, if their fibered rotation number is diophantine or rational with respect to the frequency, such cocycles are in fact reducible. This extends Eliasson's theorem on Schr\"odinger cocycles to the differentiable case

Spectral Analysis of GRBs Measured by RHESSI

The Ge spectrometer of the RHESSI satellite is sensitive to Gamma Ray Bursts (GRBs) from about 40 keV up to 17 MeV, thus ideally complementing the Swift/BAT instrument whose sensitivity decreases above 150 keV. We present preliminary results of spectral fits of RHESSI GRB data. After describing our method, the RHESSI results are discussed and compared with Swift and Konus.Comment: 4 pages, 4 figures, conference proceedings, 'Swift and GRBs: Unveiling the Relativistic Universe', San Servolo, Venice, 5-9 June 2006, to appear in Il Nouvo Ciment

Distribution of periodic points of polynomial diffeomorphisms of C^2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of \C^2: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$

Motion of vortices implies chaos in Bohmian mechanics

Bohmian mechanics is a causal interpretation of quantum mechanics in which particles describe trajectories guided by the wave function. The dynamics in the vicinity of nodes of the wave function, usually called vortices, is regular if they are at rest. However, vortices generically move during time evolution of the system. We show that this movement is the origin of chaotic behavior of quantum trajectories. As an example, our general result is illustrated numerically in the two-dimensional isotropic harmonic oscillator.Comment: 7 pages 5 figure

Aortic calcification and femoral bone density are independently associated with left ventricular mass in patients with chronic kidney disease

Background Vascular calcification and reduced bone density are prevalent in chronic kidney disease and linked to increased cardiovascular risk. The mechanism is unknown. We assessed the relationship between vascular calcification, femoral bone density and left ventricular mass in patients with stage 3 non-diabetic chronic kidney disease in a cross-sectional observational study. Methodology and Principal Findings A total of 120 patients were recruited (54% male, mean age 55±14 years, mean glomerular filtration rate 50±13 ml/min/1.73 m2). Abdominal aortic calcification was assessed using lateral lumbar spine radiography and was present in 48%. Mean femoral Z-score measured using dual energy x-ray absorptiometry was 0.60±1.06. Cardiovascular magnetic resonance imaging was used to determine left ventricular mass. One patient had left ventricular hypertrophy. Subjects with aortic calcification had higher left ventricular mass compared to those without (56±16 vs. 48±12 g/m2, P = 0.002), as did patients with femoral Z-scores below zero (56±15 vs. 49±13 g/m2, P = 0.01). In univariate analysis presence of aortic calcification correlated with left ventricular mass (r = 0.32, P = 0.001); mean femoral Z-score inversely correlated with left ventricular mass (r = −0.28, P = 0.004). In a multivariate regression model that included presence of aortic calcification, mean femoral Z-score, gender and 24-hour systolic blood pressure, 46% of the variability in left ventricular mass was explained (P<0.001). Conclusions In patients with stage 3 non-diabetic chronic kidney disease, lower mean femoral Z-score and presence of aortic calcification are independently associated with increased left ventricular mass. Further research exploring the pathophysiology that underlies these relationships is warranted

Stable manifolds and homoclinic points near resonances in the restricted three-body problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem