Chapter 1. Introduction 1 Chapter 2. Heuristic discussion of the mechanism 2.1. Integrable systems, resonances, secondary tori 2.2. Heuristic description of the mechanism Chapter 3. A simple model Chapter 4. Statement of rigorous results Chapter 5. Notation and definitions, resonances Chapter 6. Geometric features of the unperturbed problem Chapter 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds 7.1. Explicit calculations of the perturbed invariant manifold Chapter 8. The dynamics inΛ ε 37 8.1. A system of coordinates forΛ ε 39 8.2. Calculation of the reduced Hamiltonian 8.3. Isolating the resonances (resonant averaging) 8.3.1. The infinitesimal equations for averaging 8.3.2. The main averaging result, Theorem 8.9 8.3.3. Proof of Theorem 8.9 8.4. The non-resonant region (KAM theorem) 8.4.1. Some results on Diophantine approximation 8.4.2. The KAM Theorem for twist maps 8.5. Analyzing the resonances 8.5.1. Resonances of order 3 and higher 8.5.2. Preliminary analysis of resonances of order one or two 8.5.3. Primary and secondary tori near the first and second order resonances 62 8.5.4. Proof of Theorem 8.30 and Corollary 8.31 8.5.5. Existence of stable and unstable manifolds of periodic orbits Chapter 9. The scattering map 9.1. Some generalities about the scattering map 9.2. The scattering map in our model: definition and computation v vi CONTENTS Chapter 10. Existence of transition chains 97 10.1. Transition chains 99 10.2. The scattering map and the transversality of heteroclinic intersections 99 10.2.1. The non-resonant region and resonances of order 3 and higher 10.2.2. Resonances of first order 10.2.3. Resonances of order 2 10.3. Existence of transition chains to objects of different topological types Chapter 11. Orbits shadowing the transition chains and proof of theorem 4.1 Chapter 12. Conclusions and remarks 12.1. The role of secondary tori and the speed of diffusion 12.2. Comparison with [DLS00] 12.3. Heuristics on the genericity properties of the hypothesis and the phenomena 12.4. The hypothesis of polynomial perturbations 12.5. Involving other objects 12.6. Variational methods 12.7. Diffusion times Chapter 13. An example Acknowledgments Bibliography

We introduce a method to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. As a first application, we consider the dynamical system given by a C r map F in a Banach space X close to a fixed point: F (x) = Ax + N (x), A linear, N (0) = 0, DN (0) = 0. We show that if X 1 is an invariant subspace of A and A satisfies certain spectral properties, then there exists a unique C r manifold which is invariant under F and tangent to X 1. When X 1 corresponds to spectral subspaces associated to sets of the spectrum contained in disks around the origin or their complement, we recover the classical (strong) (un)stable manifold theorems. Our theorems, however, apply to other invariant spaces. Indeed, we do not require X 1 to be an spectral subspace or even to have a complement invariant under A.

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.

Given a normally hyperbolic invariant manifold Λ for a map f , whose stable and unstable invariant manifolds intersect transversally, we consider its associated scattering map. That is, the map that, given an asymptotic orbit in the past, gives the asymptotic orbit in the future. We show that when f and Λ are symplectic (resp. exact symplectic) then, the scattering map is symplectic (resp. exact symplectic). Furthermore, we show that, in the exact symplectic case, there are extremely easy formulas for the primitive function, which have a variational interpretation as difference of actions. We use this geometric information to obtain efficient perturbative calculations of the scattering map using deformation theory. This perturbation theory generalizes and extends several results already obtained using the Melnikov method. Analogous results are true for Hamiltonian flows. The proofs are obtained by geometric natural methods and do not involve the use of particular coordinate systems, hence the results can be used to obtain intersection properties of objects of any type. We also reexamine the calculation of the scattering map in a geodesic flow perturbed by a quasi-periodic potential. We show that the geometric theory reproduces the results obtained in [DLS06b] using methods of fast-slow systems. Moreover, the geometric theory allows to compute perturbatively the dependence on the slow variables, which does not seem to be accessible to the previous methods.

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