R. de la Llave

KAM theory without action-angle variables

We give a proof of a KAM theorem on existence of invariant tori with a Diophantine rotation vector for Hamiltonian systems. The method of proof is based on the use of the geometric properties of Hamiltonian systems which, in particular, do not require the Hamiltonian system either to be written in action-angle variables or to be a perturbation of an integrable one. The proposed method is also useful to compute numerically invariant tori for Hamiltonian systems. We also prove a translated torus theorem in any number of degrees of freedom.

Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems

We give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous, then it is as smooth as the maps. We furthermore prove generalizations of these facts for non-uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions.

Accurate Strategies for K.A.M. Bounds and Their Implementation

We study perturbative expansions for quasi-periodic solutions of non—linear systems. We describe how to construct and implement algorithms that prove convergence of these expansions for values of the perturbation parameter as close to optimal as desired. The method is based on a constructive form of K.A.M. theory and implemented using interval arithmetic. For some cases, the algorithms have been run on a computer yielding results better than 90% of optimal.

Manifolds on the verge of a hyperbolicity breakdown.

We study numerically the disappearance of normally hyperbolic invariant tori in quasiperiodic systems and identify a scenario for their breakdown. In this scenario, the breakdown happens because two invariant directions of the transversal dynamics come close to each other, losing their regularity. On the other hand, the Lyapunov multipliers associated with the invariant directions remain more or less constant. We identify notable quantitative regularities in this scenario, namely that the minimum angle between the two invariant directions and the Lyapunov multipliers have power law dependence with the parameters. The exponents of the power laws seem to be universal.

Analytic regularity of solutions of Livsic's cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems

We study Livsics problem of nding satisfying X = where is a given function and X is a given Anosov vector eld. We show that, if is a continuous solution and X; are analytic, then is analytic. We use the previous result to show that if two low-dimensional Anosov systems are topologically conjugate and the Lyapunov exponents at corresponding periodic points agree, the conjugacy is analytic. Analogous results hold for diieomorphisms.

Gravity Waves on the Surface of the Sphere

SummaryWe propose a Hamiltonian model for gravity waves on the surface of a fluid layer surrounding a gravitating sphere. The general equations of motion are nonlocal and can be used as a starting point for simpler models, which can be derived systematically by expanding the Hamiltonian in dimensionless parameters. In this paper, we focus on the small wave amplitude regime. The first-order nonlinear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several finite-dimensional subspaces on which the motion is integrable.

Further rigidity properties of conformal Anosov systems

We consider systems that have some hyperbolicity behavior and which preserve conformal structures on the stable and unstable bundles. We show that two such systems that are topologically conjugate are smoothly conjugate. This is somewhat more general than a conjecture of the author in 2002. Related results have also been obtained by B. Kalinin and V. Sadovskaia.

Transport in a slowly perturbed convective cell flow

We study transport properties in a simple model of two-dimensional roll convection under a slow periodic (period of order 1/ varepsilon >>1) perturbation. The problem is considered in terms of conservation of the adiabatic invariant. It is shown that the adiabatic invariant is well conserved in the system. It results in almost regular dynamics on large time scales (of order approximately varepsilon (-3) ln varepsilon ) and hence, fast transport. We study both generic systems and an example having some symmetry. (c) 2002 American Institute of Physics.

Singularity theory for non-twist KAM tori

In this monograph the authors introduce a new method to study bifurcations of KAM tori with fixed Diophantine frequency in parameter-dependent Hamiltonian systems. It is based on Singularity Theory of critical points of a real-valued function which the authors call the potential. The potential is constructed in such a way that: nondegenerate critical points of the potential correspond to twist invariant tori (i.e. with nondegenerate torsion) and degenerate critical points of the potential correspond to non-twist invariant tori. Hence, bifurcating points correspond to non-twist tori.

On the integrability of intermediate distributions for Anosov diffeomorphisms

We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C ∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.