Introduction to Dynamical Cohomology

Professor: Alejandro Kocsard
Dates:  September 16 - 27, 2013

Program:

Cohomology Equations: [KH96], [Kat01], [Koc12].

1. Generalities. "Naive" resolution method.
2. Gotschalk-Hedlund Theorem.
3. Cohomology Obstructions: Invariants measures and distributions.
4. Comology stability.

Cohomology (real) of hyperbolic systems: [KH96], [dlLMM86], [Qua97].

1. Livshitz theorem (Holder regularity)
2. Llave-Marco-Moriyon theorem.
3. Some results about rigidity of hyperbolic systems.

Cohomology of elliptic systems and distributional unique ergodic systems: [Kat01], [Hur01], [For08], [Koc09], [AK11], [AFK12].

1. Cohomology equations for translations in the torus.
2. Katok-Herman conjecture about classification of cohomology rigid systems.
3. "Exotics" examples of DUE systems.

Dynamic Cohomology with coefficients in non-commutative groups: [Kat01], [Kal11].

References:

[AFK12] Artur Avila, Bassam Fayad, and Alejandro Kocsard, Distributionally
 uniquely ergodic diffeomorphisms, Preprint, 2012.
[AK11] Artur Avila and Alejandro Kocsard, Cohomological equations and
invariant distributions for minimal circle diffeomorphisms, Duke Math. J. 158 (2011), no. 3, 501-536. MR 2805066
[dlLMM86] R. de la Llave, J. M. Marco, and R. Moriyón, Canonical perturbation theory of anosov systems and regularity results for the livsic cohomology equation, Annals of Mathematics 123 (1986), no. 3,
537-611. MR MR840722 (88h:58091)