# Sergei Yakovenko's blog: on Math and Teaching

## Finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$, I. Formal theory.

1. Formal normal form for a single holomorphic self-map from $\text{Diff}(\mathbb C^1,0)$. Parabolic germs.
2. Bochner theorem on holomorphic linearization of finite groups.
3. Stratification of the subgroup of parabolic germs $\text{Diff}_1(\mathbb C^1,0)$.
4. Tits alternative for finitely generated subgroups of $\text{Diff}(\mathbb C^1,0)$: every such subgroup is either metabelian (its commutator is commutative, e.g., trivial), or non-solvable (all iterated commutators are nontrivial).
5. Centralizers and symmetries: formal classification of solvable subgroups.
6. Integrable germs and their holomorphic linearizability.

Recommended reading: Section 6 (first part) from the book (printing disabled)

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## Wednesday, November 28, 2007

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

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 9:01
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## 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.

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