# Sergei Yakovenko's blog: on Math and Teaching

## Tuesday, November 18, 2014

### Lecture 5 (Nov. 17)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 12:27
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## Fuchsian equivalence and Fuchsian classification

Definitions
A (formal, genuine) Fuchsian operator of order $n$ is a (formal, resp., converging) series of the form $L=\sum_{k=0}^\infty t_kp_k(\epsilon)$ with the coefficients $p_k\in\mathbb C[\epsilon]$ from the ring of polynomials in the variable $\epsilon$ with $n=\deg p_0\ge \deg p_k\ \forall k=1,2,\dots$.
The polynomial $p_0\in\mathbb C[\epsilon]$ is called the Euler part of $L$.

Two Fuchsian operators $L,M$ are $\mathscr F$-equivalent (formally or analytically), if there exist two Fuchsian operators $H,K$ such that $MH=KL$ and the Euler parts of $H,L$ are mutually prime in $\mathbb C[\epsilon]$.

Unlike the Weyl algebra $\mathscr M(\mathbb C,0)[\epsilon]$, the collection of Fuchsian operators is not a subalgebra, although it is “multiplicatively” (compositionally) closed.

The Fuchsian equivalence is indeed reflexive, transitive and symmetric. The first two properties are obvious, to prove the last one an additional effort is required. Indeed, for two Fuchsian operators $L,H$ of order $n$ with mutually prime Euler parts, one can construct two operators $U,V$ with holomorphic coefficients so that the identity $UL+VH=1$ holds, but in the leading terms of $U,V$ may well degenerate, thus violating the Fuchsian condition. However, one can always find such pair of operators of order greater by 1, which will still be Fuchsian. The rest is easy.

The following results can be proved by more or less direct computation in the algebra $\mathscr W=\mathbb C[[t]]\otimes\mathbb C[\epsilon]$:

• A Fuchsian operator with a nonresonant Euler part is $\mathscr F$-equivalent to its Euler part.
• Any Fuchsian operator is $\mathscr F$-equivalent to a polynomial operator from $\mathbb C[t]\otimes\mathbb C[\epsilon]$.
• Any Fuchsian operator is $\mathscr F$-equivalent to a polynomial operator of the form $L=(\epsilon -\lambda_1+q_1(t))\cdots(\epsilon -\lambda_n+q_n(t))$ with $q_1(t),\dots,q_n(t)$ being polynomials without free terms, $q_i(0)=0$, which is Liouville integrable.
• A Fuchsian operator has trivial (identical) monodromy if and only if it is $\mathscr F$-equivalent to an Euler operator with pairwise different integer roots. The corresponding equation has an apparent singularity (all solutions are analytic) if and only if all these roots are pairwise different nonnegative integers.
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## Local theory of Fuchsian systems (cont.)

• Resonant normal form.
Definition. A meromorphic Fuchsian singularity $\dot X=t^{-1}(A_0+tA_1+\cdots+t^k A_k+\cdots)X$, $A_0=\mathrm{diag}(\lambda_1,\dots,\lambda_n)+\mathrm N$, is in the (Poincare-Dulac) normal form, if for all $k=1,2,\dots$, the identities $t^\Lambda A_k t^{-\Lambda}=t^k A_k$ hold.
• Theorem. Any Fuchsian system is holomorphically gauge equivalent to a system in the normal form.
• Integrability of the normal form: let $I=\mathrm N+A_1+\cdots +A_k+\cdots$ (in fact, the sum is finite). Then the solution is given by the (non-commutative) product $X(t)=t^\Lambda t^I$. The monodromy is the (commutative) product, $M=\mathrm e^{2\pi \mathrm i \Lambda}\mathrm e^{2\pi\mathrm i I}$.

References: [IY], section 16.

## Linear high order homogeneous differential equations

• Differential operators as noncommutative polynomials in the variable $\partial=\frac {\mathrm d}{\mathrm dt}$ with coefficients in a differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs at the origin.
• Composition and factorization.
• Reduction of a linear equation $Lu=0$ to a system of linear first order equations and back. Singular and nonsingular equations.
• Euler derivation $\epsilon=t\partial$ and Fuchsian equations (“nonsingular with respect to $\epsilon$“).
• Division with remainder, greatest common divisor of two operators, divisibility and common solutions of two equations.
• Sauvage theorem. Tame equations are Fuchsian.

References: [IY], Section 19.

# 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:

## 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.

Disclaimer is as sadly relevant as before…

## 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.

Disclaimer, alas, is still relevant…

## Wednesday, November 21, 2007

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

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 5:41
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## 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) .

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