Lecture 9 (Thu, Apr 17, 2008)

Irregular singularities of linear systems

1. One-dimensional case: complete classification.
2. Polynomial “normal forms”: Birkhoff theorem and its “uselessness”.
3. Local reducibility: similarities and differences with the regular (Fuchsian) case.
4. Polynomial “normal form” for irreducible irregular singularity: Bolibruch theorem
5. First steps of the “genuine” normal forms theory.
   • Resonances.
   • Formal diagonalizability of nonresonant systems
   • Divergence of the normalizing transformations

Recommended reading: Section 20 from the Book

Notice

The next week there will be no classes for this reason. Expect the end of the story on May 1, 2008. In the meantime I wish to everybody חג פסח שמח and merry holidays.

Recommended reading:

“Auxiliary Lesson” שעור עזר) #7) December 6, 2007

Invariant manifolds for hyperbolic maps. Complex hyperbolicity.

1. Formal theory: cross-resonances.
2. Hadamard-Perron theorem for holomorphisms. Contracting map principle reactivated.
3. Hadamard-Perron theorem for vector fields. Complex hyperbolicity.
4. Invariant hypernolic curve for saddle-nodes.
5. Poincare resonances.
6. Center manifolds: formal but non-analytic.

Reading: Section 7 from the book (printing disabled), Section 27 (parts A-C) from the book (printing disabled)

Disclaimer is as sadly relevant as before…

“Auxiliary Lesson” שעור עזר) #6) November 29, 2007

Holomorphic normalization

1. Poincaré and Siegel domains. Different types of resonances.
2. Fixed point equation and its linearization.
3. Invertibility of the homological operator.
4. Majorant norm and its properties.
5. Poincaré theorem on holomorphic linearization of vector fields in the Poincaré domain.
6. Further results: Poncare-Dulac polynomial normal form in the Poincare domain. Siegel and Brjuno theorems. Yoccoz counterexample. Divergence dychotomy.
7. Normal forms of the self-maps. Schröder-Kœnigs theorem.

Disclaimer, alas, is still relevant…

Reading: Section 5 from the book (printing disabled).

“Auxiliary Lesson” שעור עזר) #5) November 22, 2007

Formal linearization and obstructions. Poincare theorem

1. Formal equivalence of formal vector fields (total recall)
2. (Additive) Resonances
3. Poincare formal linearization theorem
4. Proof of the Poincare theorem:
   • Homological equation
   • Commutator with diagonal linear vector field
   • Stabilization of the series
5. Resonant monomials. Resonant normal form. Poincare–Dulac paradigm.
6. Formal classification of formal self-maps. Multiplicative resonances.
7. Survey of further results. Formal types of line and planar singularities.

Reading material: Section 4 from the book (printing disabled).Disclaimer (alas, still required) .