On the enhancement of diffusion by chaos, escape rates and stochastic instability

We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.

     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

On the uniform hyperbolicity of some nonuniformly hyperbolic systems

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability (probability one with respect to every invariant probability measure) are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.

     

Dynamics of horizontal-like maps in higher dimension

We study the regularity of the Green currents and of the equilibrium measure associated to a horizontal-like map in Ck, under a natural assumption on the dynamical degrees. We estimate the speed of convergence towards the Green currents, the decay of correlations for the equilibrium measure and the Lyapounov exponents. We show in particular that the equilibrium measure is hyperbolic. We also show that the Green currents are the unique invariant vertical and horizontal positive closed currents. The results apply, in particular, to Hénon-like maps, to regular polynomial automorphisms of Ck and to their small perturbations.     

Nonlinear perturbations of nonuniform exponential dichotomy on measure chains

This paper focuses on nonlinear perturbations of flows in Banach spaces, corresponding to a nonautonomous dynamical system on measure chains admitting a nonuniform exponential dichotomy. We first define the nonuniform exponential dichotomy of linear nonuniformly hyperbolic systems on measure chains, then establish a new version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics on measure chains with the help of nonuniform exponential dichotomies. Moreover, we also construct stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy. In particular, it is shown that the stable invariant manifolds are Lipschitz in the initial values provided that the nonlinear perturbation is a sufficiently small Lipschitz perturbation.     

Conjugacies and invariant manifolds via evolution semigroups

For a linear evolution family with a nonuniformly hyperbolic behavior, we give simple proofs of the existence of topological conjugacies and stable invariant manifolds under sufficiently small perturbations. The proofs are obtained via evolution semigroups, which allows us to pass from the original nonautonomous nonuni-formly hyperbolic dynamics to one that is autonomous and uniformly hyperbolic.     

Conservation of hyperbolic tori in Hamiltonian systems

In 2000, Bolotin and Treshchev proposed an invariant definition of the hyperbolic torus, generalizing the traditional coordinate definition. Simultaneously, they conjectured that, under standard assumptions on its Diophantine properties, nondegeneracy, and analyticity, the hyperbolic torus is conserved in the case of small perturbations. This conjecture generalizes Graff’s theorem. In the present paper, this conjecture is shown to be valid.     

On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure of a Holder potential , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of . If the measure of maximal entropy μ ₀ is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if is 1-sided Bernoulli for f expanding, then must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.     

Stably ergodic diffeomorphisms which are not partially hyperbolic

We show stable ergodicity of a class of conservative diffeomorphisms ofT ⁿ which do not have any hyperbolic invariant subbundle. Moreover, the uniqueness of SRB (Sinai-Ruelle-Bowen) measure for non-conservativeC ¹ perturbations of such diffeomorphisms is verified. This class strictly contains non-partially hyperbolic robustly transitive diffeomorphisms constructed by Bonatti-Viana [4] and so we answer the question posed there on the stable ergodicity of such systems.     

Hyperbolic submodels of an incompressible viscoelastic Maxwell medium

A two-dimensional motion of an incompressible viscoelastic Maxwell continuum is considered. The system of quasilinear equations describing this motion has both real and complex characteristics. A class of effectively one-dimensionalmotions is analyzed for which the original system of equations is decomposed into a hyperbolic subsystem and a quadrature. The properties of the hyperbolic submodels obtained depend on the choice of the invariant derivative in the rheological relation. When one chooses the Jaumann corotational derivative as the invariant derivative, the equations of the submodel remain quasilinear. They can be represented in the form of conservation laws, which allows one to analyze discontinuous solutions to these equations. When one chooses the upper or lower convected derivative, the equations of one-dimensional hyperbolic submodels turn out to be linear. The problem of shear motion between parallel plates and the problem of interaction between the stress field that does not depend on one of the coordinates and a transverse shear flow with initially constant vorticity are studied in detail. It is established that a plane Couette flow in the model with the corotational derivative is unstable in the linear approximation in the class of shear flows if the Weissenberg number is greater than one. The development of small perturbations gives rise to discontinuities in tangential velocities and stresses. The hysteresis phenomenon is observed when the Weissenberg number successively increases and decreases while passing through a critical value. The Couette flow in models with the upper and lower convected derivative remains stable with respect to one-dimensional perturbations.     

A Nonholonomic Model of the Paul Trap

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.     

SRB measures for partially hyperbolic systems whose central direction is mostly contracting

We consider partially hyperbolic diffeomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundleE ^uu (uniformly expanding) and a subbundleE ^c, dominated byE ^uu. We prove that if the central directionE ^c is mostly contracting for the diffeomorphism (negative Lyapunov exponents), then the ergodic Gibbsu-states are the Sinai-Ruelle-Bowen measures, there are finitely many of them, and their basins cover a full measure subset. If the strong-unstable leaves are dense, there is a unique Sinai-Ruelle-Bowen measure. We describe some applications of these results, and we also introduce a construction of robustly transitive diffeomorphisms in dimension larger than three, having no uniformly hyperbolic (neither contracting nor expanding) invariant subbundles. Work partially supported by CNRS and CNPq/PRONEX-Dynamical Systems, and carried out at Laboratoire de Topologie, Dijon, and IMPA, Rio de Janeiro.     

Persistence of lower-dimensional hyperbolic invariant tori for generalized Hamiltonian systems

In this paper we study the persistence of lower dimensional hyperbolic invariant tori for generalized Hamiltonian systems. Here the generalized Hamiltonian systems refer to the systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd-dimensional. Under Rüssmann-type non-degenerate condition, by introducing a modified linear KAM iterative scheme, we proved that the majority of the lower-dimensional hyperbolic invariant tori persist under small perturbations for generalized Hamiltonian systems.     

Invariant hyperbolic tori for Gevrey-smooth Hamiltonian systems under Rüssmann’s non-degeneracy condition

In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann＇s nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian.     

Recurrence times and rates of mixing

The setting of this paper consists of a map making “nice” returns to a reference set. Criteria for the existence of equilibria, speed of convergence to equilibria and for the central limit theorem are given in terms of the tail of the return time function. The abstract setting considered arises naturally in differentiable dynamical systems with some expanding or hyperbolic properties. The author is partially supported by a grant from the National Science Foundation and a Guggenheim Fellowship.     

Exponential inequalities for dynamical measures of expanding maps of the interval

 We prove an exponential inequality for the absolutely continuous invariant measure of a piecewise expanding map of the interval. As an immediate corollary we obtain a concentration inequality. We apply these results to the estimation of the rate of convergence of the empirical measure in various metrics and also to the efficiency of the shadowing by sets of positive measure. Received: 14 August 2001 / Revised version: 13 February 2002 / Published online: 1 July 2002     

A Compactification of the Space of Expanding Maps on the Circle

We show the space of expanding Blaschke products on S¹ is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure preserving topological covering maps of S¹. Research supported in part by the NSF.     

Convergence of a misanthrope process to the entropy solution of 1D problems

We prove the convergence, in some strong sense, of a Markov process called “a misanthrope process” to the entropy weak solution of a one-dimensional scalar nonlinear hyperbolic equation. Such a process may be used for the simulation of traffic flows. The convergence proof relies on the uniqueness of entropy Young measure solutions to the nonlinear hyperbolic equation, which holds for both the bounded and the unbounded cases. In the unbounded case, we also prove an error estimate. Finally, numerical results show how this convergence result may be understood in practical cases.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.