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Syllabus 2013



Barcelona, July 15 - 19, 2013

Course Syllabus
Quasiperiodic solutios in Hamiltonian dynamical systems and the Hamilton-Jacobi equation, with applications to Celestial Mechanics

Jaques Féjoz (Université Paris-Dauphine)
  • Hamiltonian systems.
  • Quasiperiodic motions.
  • A more geometric viewpoint.
  • The Hamilton-Jacobi equation.
  • Persistence of invariant tori.
  • Application to the three-body problem.
Robustly transitive dynamics 

Enrique Pujals (IMPA, Rio de Janeiro)
  • Robust transitivity.
  • Examples of robust transitive partially hyperbolic systems.
  • Dominated splitting: examples of robust transitive systems which are not
    partially hyperbolic.
  • Dynamical consequences from robust transitivity.
  • Classi cation of partially hyperbolic systems.
  • Blenders and Iterated Function Systems.
  • Partially hyperbolic systems in the Symplectic and Hamiltonian context.
  • Density of homoclinic and heteroclinic bifurcations in the complement of
    hyperbolic systems.


  • References

[BD] C. Bonatti, L. J. Diaz, Persistence of transitive di eomorphisms, Annals of Math 143
(1995), 367-396.
[BDP] C. Bonatti, L. J. Diaz, E. R. Pujals, A C1-generic dichotomy for di eomorphisms:
weak form of hyperbolicity or in nitely many sinks or sources, Annals of Mathematics,
158 (2003), 355-418.
[BDU] C. Bonatti, L.J. Daz, R. Ures; Minimality of strong stable and unstable foliations for
partially hyperbolic di eomorphisms. J. Inst. Math. Jussieu 1 (2002), 4, 513{541.
[BFP] J. Bochi, B. Fayad, E. R. Pujals, Dichotomy for conservative robust ergodic maps, to
appear in CRAS.
[CP] S. Crovissier, E. R. Pujals; Essential hyperbolicity and homoclinic bifurcations: a
dichotomy phenomenon/mechanism for di eomorphisms.
[D1] L. J. Diaz Robust nonhyperbolic dynamics at heterodimensional cycles. Ergodic The-
ory and Dynamical Systems, 15 (1995), 291-315.

[HPS] M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Springer Lecture Notes in Math.,
583 (1977).
[LP] C. Lizana, E. R. Pujals, Robust transivity for Endomorphisms, to appear in Ergodic
Theory and Dynamical Systems.
[M] R. Ma~ne, Contributions to the stability conjecture. Topology, 17 (1978), 386-396.
[M2] R. Ma~ne, An ergodic clossing lemma, Ann. of Math. 116 (1982), 503-540.
[NP] M. Nassiri, E. R. Pujals,Robust transivity in Hamiltonian Dynamics, Annales scien-
ti ques de l'Ecole Mormale Superieur (45), fascicule 2 (2012), 191-239.
[PS1] E. R. Pujals, M. Sambarino,Topics on homoclinic bifurcation, dominated splitting,
robust transitivity and related results, Handbook of dynamical systems vol 1B, Elsevier
(2005) 327-378
[PS2] E. R. Pujals, M. Sambarino, Homoclinic tangencies and hyperbolicity for surface
di eomorphisms, Annals of Mathematics, 151 (2000), 961-1023.
[PS3] E. R. Pujals, M. Sambarino, On the dynamics of dominated splitting, Annals of
Mathematics (169) (2009), 675-740.

    Regularity of the free boundary in problems with distributed sources

    Sandro Salsa (Politecnico di Milano)
    • One phase free boundary problems:
    1. a Harnack inequality
    2. flat free boundaries are smooth
    3. Lipschitz free boundaries are amooth
    • Two phase f.b.p's:
    1. nondegeneracy properties
    2. a Harnack inequality
    3. regularity in transmission problem
    4. improvement of flatness: the non-degenerate case
    5. the degenerate case
    6. a loiuville theorem and the smoothness of Lipschitz free boundary
    Regularity results for nonlocal equations

    Luis Silvestre (University of Chicago)
    • Lecture 1
      • 1.1 Definitions: linear equations
      • 1.2 Probabilistic derivation
      • 1.3 Uniform ellipticity
    • Lecture 2
      • 2.1 Viscosity solutions
      • 2.2 An open problem
      • 2.3 Second order equations as limits of integro-differential equations
      • 2.4 Smooth approximations of viscosity solutions to fully nonlinear elliptic equations
      • 2.5 Regularity of nonlinear equations: how to start
      • 2.5.1 Differentiating the equation
      • 2.5.2 Holder estimates
    • Lecture 3
        • 3.1 C1,? estimates for nonlinear nonlocal equations
        • 3.2 Holder estimates in the parabolic case
    • Lecture 4
        • 4.1 How to prove Holder estimates for 2nd order equations
        • 4.1.1 The ABP estimate
        • 4.1.2 A special bump function
        • 4.1.3 A first estimate in measure
        • 4.1.4 The Lε estimate
        • 4.2 About the proof of uniform Holder estimates for non local equations
      • Lecture 5
          • 5.1 The (classical?) second order case: the Evans-Krylov theorem
          • 5.1.1 The assumptions revisited
          • 5.1.2 The half Harnack inequalities
          • 5.1.3 The a priori estimate in C1,1
          • 5.1.4 The a priori estimate in C2,α
          • 5.2 The integro-differential Bellman equation
          • 5.2.1 Reinterpreting the assumptions
          • 5.2.2 The estimate in Ws,? (what that means)
          • 5.2.3 The estimate in Cs+α
          • References