On the Lebesgue Measure of Li-Yorke Pairs for Interval Maps

We investigate the prevalence of Li-Yorke pairs for C ² and C ³ multimodal maps f with non-flat critical points. We show that every measurable scrambled set has zero Lebesgue measure and that all strongly wandering sets have zero Lebesgue measure, as does the set of pairs of asymptotic (but not asymptotically periodic) points.     

Global attractors and invariant measures for non-invertible planar piecewise isometric maps

The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A.     

The Navier-Stokes equations on a bounded domain

SupposeU is an open bounded subset of 3-space such that the boundary ofU has Lebesgue measure zero. Then for any initial condition with finite kinetic energy we can find a global (i.e. for all time) weak solutionu to the time dependent Navier-Stokes equations of incompressible fluid flow inU such that the curl ofu is continuous outside a locally closed set whose 5/3 dimensional Hausdorff measure is finite.This research was supported in part by the National Science Foundation Grant MCS-7903361     

Sensitivity to initial conditions, entropy production, and escape rate at the onset of chaos

We analytically link three properties of nonlinear dynamical systems, namely sensitivity to initial conditions, entropy production, and escape rate, in z-logistic maps for both positive and zero Lyapunov exponents. We unify these relations at chaos, where the Lyapunov exponent is positive, and at its onset, where it vanishes. Our result unifies, in particular, two already known cases, namely (i) the standard entropy rate in the presence of escape, valid for exponential functionality rates with strong chaos, and (ii) the Pesin-like identity with no escape, valid for the power-law behavior present at points such as the Feigenbaum one.     

The baker transformation and a mapping associated to the restricted three body problem

The nonlinear mapping of the plane $$\begin{array}{*{20}c} {x_1 = x_0 + 1/y_0 } \\ {y_1 = y_0 - x_0 - 1/y_0 } \\ \end{array} $$ was recently introduced by Hénon as an asymptotic form of the equations of motion of the restricted three body problem. This is an area preserving diffeomorphism, except along thex-axis where the mapping is singular. We show that this mapping exhibits a type of stochastic behavior known as topological transitivity, by showing that it is topologically conjugate to the well known baker transformation. Consequently, periodic points are dense in the plane and there is also a dense orbit. We note that the baker transformation also preserves Lebesgue measure and is ergodic, so this raises interesting open questions about the ergodic properties of the nonlinear mapping.     

On the asymptotic behavior in the scattering problem for several charged quantum particles interacting via repulsive pair potentials

The ansatz proposed earlier by Buslaev and Levin [Funct. Anal. Appl. 46, 147 (2012)] for describing the leading order in the asymptotic behavior of continuum eigenfunctions in the scattering problem for three charged quantum particles is extended to the system of n charged quantum particles of identical mass and identical charge. It is shown that the proposed ansatz generates a fast decreasing (faster than the potential) discrepancy of the Schrödinger equation for a set of asymptotic configurations.     

On the complete synchronization of the Kuramoto phase model

We discuss the asymptotic complete phase-frequency synchronization for the Kuramoto phase model with a finite size N. We present sufficient conditions for initial configurations leading to the exponential decay toward the completely synchronized states. Our new sufficient conditions and decay rate depend only on the coupling strength and the diameter of initial phase and natural frequency configurations. But they are independent of the system size N, hence they can be used for the mean-field limit. For the complete synchronization estimates, we estimate the time evolution of the phase and frequency diameters for configurations. The initial phase configurations for identical oscillators located on the half circle will converge to the complete synchronized states exponentially fast. In contrast, for the non-identical oscillators, the complete frequency synchronization will occur exponentially fast for some restricted class of initial phase configurations. Our estimates are based on the monotonicity arguments of extremal phase and frequencies, which do not employ any linearization procedure of nonlinear coupling terms and detailed information on the eigenvalue of the linearized system around the complete synchronized states. We compare our analytical results with numerical simulations.     

A Low Temperature Analysis of the Boundary Driven Kawasaki Process

Low temperature analysis of nonequilibrium systems requires finding the states with the longest lifetime and that are most accessible from other states. We determine these dominant states for a one-dimensional diffusive lattice gas subject to exclusion and with nearest neighbor interaction. They do not correspond to lowest energy configurations even though the particle current tends to zero as the temperature reaches zero. That is because the dynamical activity that sets the effective time scale, also goes to zero with temperature. The result is a non-trivial asymptotic phase diagram, which crucially depends on the interaction coupling and the relative chemical potentials of the reservoirs.     

Spectral properties of one dimensional quasi-crystals

In this paper we prove that the one dimensional Schrödinger operator onl ²(?) with potential given by: $$\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha \notin \mathbb{Q}$$ has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all .     

Non-Stationary Non-Uniform Hyperbolicity: SRB Measures for Dissipative Maps

We prove the existence of SRB measures for diffeomorphisms where a positive volume set of initial conditions satisfy an “effective hyperbolicity” condition that guarantees certain recurrence conditions on the iterates of Lebesgue measure. We give examples of systems that do not admit a dominated splitting but can be shown to have SRB measures using our methods.     

Numerical experiments of the planar equal-mass three-body problem: the effects of rotation

The planar three-body problem with angular momentum is numerically and systematically studied as a generalization of the free-fall problem (i.e., the three-body problem with zero initial velocities). The initial conditions in the configuration space exhaust all possible forms of a triangle, whereas the initial conditions in the momentum space are chosen so that position vectors and momentum vectors are orthogonal. Numerical results are organized according to the value of virial ratio k defined as the ratio of the total kinetic energy to the total potential energy. Final motions are mapped in the initial value space. Several interesting features are found. Among others, binary collision curves seem to spiral into the Lagrange point, and for large k, binary collision curves connect the Lagrange point and the Euler point. The existence of a lunar periodic orbit and a periodic orbit of petal-type is suggested. The number of escape orbits as a function of the escape time is analyzed for different k. The behavior of this number for different time and k shows most remarkably the effects of rotation of triple systems. The number of escape orbits increases exponentially for k相似文献   

Scattering theory in the energy space for a class of non-linear wave equations

We study the asymptotic behaviour in time of the solutions and the theory of scattering in the energy space for the non-linear wave equation $$\square \varphi + f(\varphi ) = 0$$ in ? ⁿ ,n≧3. We prove the existence of the wave operators, asymptotic completeness for small initial data and, forn≧4, asymptotic completeness for arbitrarily large data. The assumptions onf cover the case wheref behaves slightly better than a single powerp=1+4/(n?2), both near zero and at infinity (see (1.5), (1.6) and (1.8)).     

Large Deviations for Non-Uniformly Expanding Maps

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved. As easy by-products we deduce escape rates from subsets of the basins of physical measures for these types of maps. The rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states. 2000 Mathematics Subject Classification: 37D25, 37A50, 37B40, 37C40     

On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.     

Radial asymptotics of Lemaître–Tolman–Bondi dust models

We examine the radial asymptotic behavior of spherically symmetric Lemaître–Tolman–Bondi dust models by looking at their covariant scalars along radial rays, which are spacelike geodesics parametrized by proper length ?, orthogonal to the 4-velocity and to the orbits of SO(3). By introducing quasi-local scalars defined as integral functions along the rays, we obtain a complete and covariant representation of the models, leading to an initial value parametrization in which all scalars can be given by scaling laws depending on two metric scale factors and two basic initial value functions. Considering regular “open” LTB models whose space slices allow for a diverging ?, we provide the conditions on the radial coordinate so that its asymptotic limit corresponds to the limit as ? → ∞. The “asymptotic state” is then defined as this limit, together with asymptotic series expansion around it, evaluated for all metric functions, covariant scalars (local and quasi-local) and their fluctuations. By looking at different sets of initial conditions, we examine and classify the asymptotic states of parabolic, hyperbolic and open elliptic models admitting a symmetry center. We show that in the radial direction the models can be asymptotic to any one of the following spacetimes: FLRW dust cosmologies with zero or negative spatial curvature, sections of Minkowski flat space (including Milne’s space), sections of the Schwarzschild–Kruskal manifold or self-similar dust solutions.     

Shell effects in the evolution of the mass asymmetry in heavy-ion collisions leading to composite systems withZ=108

Fragment mass distributions are presented obtained in the heavy-ion reactions²²Ne+²⁴⁹Cf,³²S+²³⁸U,⁴⁰Ar+²³²Th and⁵⁶Fe+²⁰⁸Pb leading to composite systems with equal nuclear charge numberZ=108. The experiments were performed at the heavy-ion cyclotron U 300 of the Laboratory of Nuclear Reactions in Dubna. The spectrometer DEMAS was used to measure the time-of-flight values and the laboratory angles of the correlated fragments. The shape of the mass distributions strongly depends on the initial mass asymmetry. When decreasing the bombarding energy down to values near the Coulomb barrier, the mass distributions obtained in the reactions³²S+²³⁸U and⁴⁰Ar+²³²Th exhibit relative maxima ofM≈205 interpreted to be due to stabilizing effects of nuclear shells during the fragmentation process.     

Typical Orbits of Quadratic Polynomials with a Neutral Fixed Point: Brjuno Type

We describe the topological behavior of typical orbits of complex quadratic polynomials ${P_{\alpha}(z) = e^{2 \pi \alpha {\bf i}} z + z^{2}}$ P α ( z ) = e 2 π α i z + z 2 , with α of high return type. Here we prove that for such Brjuno values of α the closure of the critical orbit, which is the measure theoretic attractor of the map, has zero area. Then we show that the limit set of the orbit of a typical point in the Julia set of P _α is equal to the closure of the critical orbit. Our method is based on the near parabolic renormalization of Inou-Shishikura, and a uniform optimal estimate on the derivative of the Fatou coordinate that we prove here.     

Rational Misiurewicz Maps are Rare

We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree d ≥ 2.     

Strong Time Operators Associated with Generalized Hamiltonians

Let the pair of operators, (H, T), satisfy the weak Weyl relation: $$T{\rm e}^{-itH}={\rm e}^{-itH}(T+t),$$ where H is self-adjoint and T is closed symmetric. Suppose that g is a real-valued Lebesgue measurable function on ${\mathbb {R}}$ such that ${g\in C^2(\mathbb {R}\backslash K)}$ for some closed subset ${K\subset\mathbb {R}}$ with Lebesgue measure zero. Then we can construct a closed symmetric operator D such that (g(H), D) also obeys the weak Weyl relation.