Syllabus 2013

• Hamiltonian systems.
• Quasiperiodic motions.
• A more geometric viewpoint.
• The Hamilton-Jacobi equation.
• Persistence of invariant tori.
• Application to the three-body problem.

• 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.
• Classication 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 dieomorphisms, Annals of Math 143
(1995), 367-396.
[BDP] C. Bonatti, L. J. Diaz, E. R. Pujals, A C1-generic dichotomy for dieomorphisms:
weak form of hyperbolicity or innitely 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 dieomorphisms. 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 dieomorphisms.
[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-
tiques 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
dieomorphisms, 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.

• 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

• 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