Transversal homoclinic points and hyperbolic sets for non-autonomous maps I

A concept of generalized hyperbolic sets for non-autonomous maps is developed. Starting from transversal homoclinic orbits such generalized hyperbolic sets are constructed. The Shadowing Lemma is proven for maps admitting a generalized hyperbolic set. Time dependent symbolic dynamics is introduced and related to non-autonomous maps.

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Zusammenfassung Das Konzept von verallgemeinerten hyperbolischen Mengen für nicht-autonome Abbildungen wird entwickelt. Ausgehend von transversalen homoklinen Bahnen werden solche verallgemeinerte hyperbolische Mengen konstruiert. Das Shadowing Lemma wird für Abbildungen bewiesen, welche eine verallgemeinerte hyperbolische Menge haben. Es wird zeitabhängige symbolische Dynamik eingeführt und der Zusammenhang mit nicht-autonomen Abbildungen dargestellt.

     

Classification of escaping exponential maps

We give a complete classification of the set of parameters for which the singular value of escapes to under iteration. In particular, we show that every path-connected component of this set is a curve to infinity.

     

Instability of exponential Collet-Eckmann maps

Given λ ∈ ℂ \ {0} let the entire function f _λ: ℂ → ℂ be defined by the formula

Transversal homoclinic points and hyperbolic sets for non-autonomous maps II

The method of Melnikov is generalized to non-autonomous maps. If the Melnikov function has infinitely many zeros with derivatives bounded away from zero then the system admits a generalized hyperbolic set as it was introduced in part I. The developed theory is applied to almost periodically perturbed differential equations.

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Zusammenfassung Die Methode von Melnikov wird verallgemeinert für nichtautonome Abbildungen. Falls die Melnikov-Funktion unendlich viele Nullstellen mit von Null weg beschränkten Ableitungen hat, dann enthält das System eine verallgemeinerte hyberbolische Menge, wie sie in Teil I eingeführt wurde. Die entwickelte Theorie wird auf fast periodisch gestörte Systeme angewandt.

     

Weak compactness of wave maps and harmonic maps

We show that a weak limit of a sequence of wave maps in (1 + 2) dimensions with uniformly bounded energy is again a wave map. Essential ingredients in the proof are Hodge structures related to harmonic maps, ¹ estimates for Jacobians, ¹-BMO duality, a “monotonicity” formula in the hyperbolic context and the concentration compactness method. Application of similar ideas in the elliptic context yields a drastically shortened proof of recent results by Bethuel on Palais-Smale sequences for the harmonic map functional on two dimensional domains and on limits of almost H-surfaces.     

Geometry and ergodic theory of non-hyperbolic exponential maps

We deal with all the maps from the exponential family such that the orbit of zero escapes to infinity sufficiently fast. In particular all the parameters are included. We introduce as our main technical devices the projection of the map to the infinite cylinder and an appropriate conformal measure . We prove that , essentially the set of points in returning infinitely often to a compact region of disjoint from the orbit of , has the Hausdorff dimension , that the -dimensional Hausdorff measure of is positive and finite, and that the -dimensional packing measure is locally infinite at each point of . We also prove the existence and uniqueness of a Borel probability -invariant ergodic measure equivalent to the conformal measure . As a byproduct of the main course of our considerations, we reprove the result obtained independently by Lyubich and Rees that the -limit set (under ) of Lebesgue almost every point in , coincides with the orbit of zero under the map . Finally we show that the the function , , is continuous.

     

Periodic structure of transversal maps on sum-free products of spheres

In this article, we study the periodic structure of transversal maps on the product of spheres of different dimensions. In particular, we give sufficient conditions in order that such maps have infinitely many even and odd periods. Moreover, we also provide sufficient conditions for having non-zero Lefschetz numbers of period m for infinitely many m's. We extend these results to transversal maps on rational exterior spaces of rank 1.     

Bifurcations in the space of exponential maps

This article investigates the parameter space of the exponential family . We prove that the boundary (in ℂ) of every hyperbolic component is a Jordan arc, as conjectured by Eremenko and Lyubich as well as Baker and Rippon. In fact, we prove the stronger statement that the exponential bifurcation locus is connected in ℂ, an analog of Douady and Hubbard’s celebrated theorem that the Mandelbrot set is connected. We show furthermore that ∞ is not accessible through any nonhyperbolic (“queer”) stable component. The main part of the argument consists of demonstrating a general “Squeezing Lemma”, which controls the structure of parameter space near infinity. We also prove a second conjecture of Eremenko and Lyubich concerning bifurcation trees of hyperbolic components. Mathematics Subject Classification (2000)  Primary 37F10, Secondary 30D05, 37F45     

On geodesic exponential maps of the Virasoro group

We study the geodesic exponential maps corresponding to Sobolev type right-invariant (weak) Riemannian metrics μ^(k) (k≥ 0) on the Virasoro group Vir and show that for k≥ 2, but not for k = 0,1, each of them defines a smooth Fréchet chart of the unital element e ∈Vir. In particular, the geodesic exponential map corresponding to the Korteweg–de Vries (KdV) equation (k = 0) is not a local diffeomorphism near the origin. A. Constantin: Supported in part by the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652). T. Kappeler: Supported in part by the SNSF, the programme SPECT, and the European Community through the FP6 Marie Curie RTN ENIGMA (MRTN-CT-2004-5652)     

On equivariant self-similar wave maps

By variational methods, we establish the existence of an infinite number of self-similar wave maps which evolve from smooth, compactly supported data and develop a singularity in finite time.     

A landing theorem for periodic rays of exponential maps

For the family of exponential maps , we show the following analog of a theorem of Douady and Hubbard concerning polynomials. Suppose that is a periodic dynamic ray of an exponential map with nonescaping singular value. Then lands at a repelling or parabolic periodic point. We also show that there are periodic dynamic rays landing at all periodic points of such an exponential map, with the exception of at most one periodic orbit.

     

Uniqueness for critical nonlinear wave equations and wave maps via the energy inequality

We show uniqueness of sufficiently regular solutions to critical semilinear wave equations and wave maps in the (a priori) much larger class of distribution solutions with finite energy, assuming only that the energy is nonincreasing in time. © 1999 John Wiley & Sons, Inc.     

Global existence for wave maps with torsion

Wave maps (i.e. nonlinear sigma models) with torsion are considered in 2+1 dimensions. Global existence of smooth solutions to the Cauchy problem is proven for certain reductions under a translation group action: invariant wave maps into general targets, and equivariant wave maps into Lie group targets. In the case of Lie group targets (i.e. chiral models), a geometrical characterization of invariant and equivariant wave maps is given in terms of a formulation using frames     

On the regularity of spherically symmetric wave maps

Wave maps are critical points U: M → N of the Lagrangian ??[U] = ∞_M ‖dU‖², where M is an Einsteinian manifold and N a Riemannian one. For the case M = ?^2,1 and U a spherically symmetric map, it is shown that the solution to the Cauchy problem for U with smooth initial data of arbitrary size is smooth for all time, provided the target manifold N satisfies the two conditions that: (1) it is either compact or there exists an orthonormal frame of smooth vectorfields on N whose structure functions are bounded; and (2) there are two constants c and C such that the smallest eigenvalue λ and the largest eigenvalue λ of the second fundamental form k_AB of any geodesic sphere Σ(p, s) of radius s centered at p ? N satisfy sλ ≧ c and s A ≦ C(1 + s). This is proved by first analyzing the energy-momentum tensor and using the second condition to show that near the first possible singularity, the energy of the solution cannot concentrate, and hence is small. One then proves that for targets satisfying the first condition, initial data of small energy imply global regularity of the solution. © 1993 John Wiley & Sons, Inc.     

Scattering for wave maps exterior to a ball

We consider 1-equivariant wave maps from ${\mathbb{R}}_t\times ({\mathbb{R}}_x^3\setminus B)\to S^3$ where $B$ is a ball centered at 0, and $\partial B$ gets mapped to a fixed point on $S^3$. We show that 1-equivariant maps of degree zero scatter to zero irrespective of their energy. For positive degrees, we prove asymptotic stability of the unique harmonic maps in the energy class determined by the degree.