Lecture 1. Theorem and mistake of Dulac Yulij Ilyashenko February - - PowerPoint PPT Presentation

Lecture 1. Theorem and mistake of Dulac

Yulij Ilyashenko February 1, 2020

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Main results

Theorem A polynomial vector field on the real plane has only finitely many limit cycles. Theorem An analytic vector field on a closed two-dimensional surface has only finitely many limit cycles. Theorem (nonaccumulation theorem) An elementary polycycle of an analytic vector field on a two-dimensional surface has a neighborhood free of limit cycles.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Small discrepancy theorem

Theorem For any n there exists an analytic vector field with a polycycle whose monodromy map has the form ∆ = id + R, R = o(exp(−exp[n]), R ≡ 0 in the log coordinate ξ = − ln x.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Semi-regular germs and Dulac’s theorem

Definition A Dulac series is a formal series of the form σ = cxν0 +

∞

• 1

Pj(ln x)xνj, where c > 0, 0 < ν0 < · · · < νj < · · · , νj → ∞, and the Pj are polynomials. Definition The germ of a mapping f : (R+, 0) → (R+, 0) is said to be semiregular if it can be expanded in an asymptotic Dulac

• series. In other words, for any N there exists a partial sum S
• f the above series such that f (x) − S(x) = o(xN).

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Examples.

1 ∆ is a C ∞ diffeo. 2 ∆(x) = xλ. 3 ∆ = h1,1 =

x 1−x ln x .

Theorem (Dulac) A monodromy map of a polycycle of an analytic vector field is either flat, or inverse to flat, or semi-regular

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Desingularization theorem

Definition An elementary singular point of a planar vector field is a point with at least one non-zero eigenvalue. Ckassification for smooth vector fields: saddles, foci, centers by linear terms, nodes, saddlenodes Theorem (Bendixson - Lefshetz - Seidenberg - Dumortier...) An isolated singular point may be split to a finite number of elementary

• nes by a finite number of blow ups. Same for C ∞ vector

fields with the Loyasievic condition: |v(x)| > c|xλ| for some C, λ > 0.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Remark An elementary monodromic polycycle consists of hyperbolic saddles and saddlenodes. Its monodromy map is a product of the Dulac maps corresponding to hyperbolic sectors of the saddles and saddlenodes. Theorem (I, 85) Dulac’s theorem holds for C ∞ vector fields whose singular points satisfy the Loyasievic condition.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

C ∞-normal forms for hyperbolic saddles

˙ x = λ1x + ..., ˙ y = λ2y + ..., λ1, λ2 > 0, λ = λ2 λ1 When λ ∈ Q, the saddle is resonant. When λ ∈ Q, the saddle is non-resonant. Theorem C ∞ orbital normal forms for a hyperbolic saddle is linear in the non-resonant case (Sternberg) integrable in the resonant case (Chen); it is either linear,

• r

˙ x = λx(1 ± uk + au2k), ˙ y = −y, u = xmy n, λ = m n .

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Hyperbolic polycycles

Remark The factor-system ˙ u = mu(±uk + au2k) is integrable. Corrolary The Dulac map of a hyperbolic saddle is semi-regular. It is xλ in the non-resonant case (in some smooth chart) Lemma Semi-regular germs form a group with respect to composition. Corrolary Dulac theorem holds for hyperbolic polycycles.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Normal forms and Dulac maps for saddlenodes

˙ x = λ1x + ..., ˙ y = λ2y + ..., λ1 = 0, λ2 = 0. Theorem (Bogdanov,...) C ∞ orbital normal forms for a saddlenode of finite multiplicity is ˙ x = ±xk+1(1 + axk)−1, ˙ y = −y, k ∈ N, a ∈ R. Corrolary The Dulac map for a saddlenode above TO the center manifold is “flat-semiregular”: ∆st = f0 ◦ hk,a, f0 = exp(−1/x), hk,a = kxk 1 − akxk ln x . The map f0 is standard flat, the map hk,a is semi-regular.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Structural Theorem 1

Theorem (Dulac) A monodromy map of an elementary polycycle is a composition of the maps of classes R, TO, FROM = TO−1. Definition The map ∆ = ∆N ◦ ... ◦ ∆1 is balanced, iff #{∆j ∈ TO} = #{∆j ∈ FROM}. Obviuosly, the non-balanced composition is either flat, or inverse to flat.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Characteristic of a composition

Definition Characteristic of a composition ∆ above is a continuous function χ on a segment [−N, 0] which is linear between two subsequent integers, χ(0) = 0, and χ(−j) = χ(−j + 1) for ∆j ∈ R, χ(−j) = χ(−j + 1) − 1 for ∆j ∈ TO, χ(−j) = χ(−j + 1) + 1 for ∆j ∈ FROM. The composition ∆ is balanced iff χ(−N) = χ(0) = 0.

Yulij Ilyashenko Lecture 1. Theorem and mistake of Dulac

Lecture 2. Finiteness theorem for polycycles with hyperbolic vertices

Theorem The limit cycles of an analytic vector field with nondegenerate singular points cannot accumulate on a polycycle of this field.

Almost regular mappings

Definition A quadratic standard domain is an arbitrary domain of the form ΩC = φC(C+), φC = ζ + C(ζ + 1)1/2, C > 0. Definition A Dulac exponential series is a formal series of the form Σ = ν0ζ + c +

• Pj(ζ) exp νjζ,

where ν0 > 0, c ∈ R, 0 > νj ց −∞, and the Pj are real polynomials; the arrow ց means monotonically decreasing convergence.

Definition An almost regular mapping is a holomorphic mapping of some quadratic standard domain Ω in C that is real on R+ and can be expanded in this domain as an asymptotic real Dulac exponential series. Expandability means that for any ν > 0 there exists a partial sum approximating the mapping to within

• (exp(−νξ)) in Ω.

Theorem An almost regular mapping is uniquely determined by an asymptotic series of it. In particular, an almost regular mapping with asymptotic series ζ is the identity. Almost regular germs at infinity form a group with the

• peration a composition.

Complexification and Phragmen-Lindel¨

• f theorem

Theorem If a function g is holomorphic and bounded in the right half-plane and decreases on (R+, ∞) more rapidly than any exponential exp(−νξ), ν > 0, then g ≡ 0. Theorem The correspondence mapping of a hyperbolic saddle, written in a logarithmic chart, extends to an almost regular mapping in some quadratic domain. This implies the finiteness theorem for hyperbolic polycycles.

(a)

Correspondence mappings for complex saddlenodes

A germ of a holomorphic vector field at an isolated degenerate elementary singular point is formally orbitally equivalent to the germ ˙ z = zk+1(1 + azk)−1, ˙ w = −w. Here k + 1 is the multiplicity of the singular point, and a is a constant that is real if the original germ is real. The Dulac map TO the center manifold has the form ∆st = exp(−1/hk,a(z)), where hk,a(z) = kzk/(1 − akzk ln z).

Monodromy of a complex saddlenode

The complexified germ of a real holomorphic vector field at an isolated degenerate elementary singular point always has a

• ne-dimensional holomorphic invariant manifold.

Corresponding monodromy transformation has the form f : z → z − 2πizk+1 + · · · . (1) This transformation is formally equivalent to a −2πi-time shift along the trajectories of the vector field v(z) = zk+1/(1 + azk). Here k and a are the same as in the formal orbital normal form of the germ.

Sectorial normalization theorem for parabolic germs

Theorem For an arbitrary parabolic germ (1) (not necessary a monodromy transformation) there exists a tuple of holomorphic functions, called a cochain normalizing the germ, having the following properties.

• 1. The functions in the tuple are in bijective correspondence

with the sectors of a nice k-partition of some disk with center zero and radius R; each function is defined in a corresponding sector.

• 2. Each function in the tuple extends biholomorphically to a

sector Sj with the same bisector and a larger angle α ∈ (π/k, 2π/k) the radius of the sector depends on α.

Theorem (Continued)

• 3. All the functions in the tuple have a common asymptotic

Taylor series at zero with linear part the identity.

• 4. In the intersections of the corresponding sectors the

functions in the tuple differ by o(exp(−c/zk)) for some c > 0.

• 5. Each of the functions in the tuple conjugates the germ (1)

in the sector Sj with the time shift by −2πi along the trajectories of the field v(z). The set of the differences in item 4 is called the coboundary of the cochaain. Cochains with these properties form the class NCk Why the coboundary decreases exponentially?

Sectorial normalization theorem for complex saddlenodes

Definition A semiformal z-preserving substitution is a substitution H of the form (z, w) → (z, w + ˆ H(z, w)), ˆ H =

∞

• 1

Hn(w)zn; the functions Hn are holomorphic in one and the same disk ; the series ˆ H of powers of z is formal (a z-preserving substitution is denoted in the same way as the correction of its second component).

Proposition For an arbitrary equation ˙ z = ±zk+1(1+azk)−1 +..., ˙ w = −w +..., k ∈ N, a ∈ R (2) there exists a unique substitution of the form h ◦ ˆ H, where ˆ H is a semiformal z-preserving substitution, and h is a holomorphic substitution of the form (z, w) → (h(z), w), h(z) − z = o(zk+1), carrying this equation into the equation ˙ z = zk+1, ˙ w = −w(1 + azk) (3)

Let D be a small disk centered at zero on the w axis. Theorem (on sectorial normalization) For an arbitrary equation (2) there exists in each sector Sj × D

• f a nice k-covering of a neighborhood of zero in C2 with the

w-axis deleted a unique biholomorphic mapping h ◦ Hj carrying equation (2) into equation (3) and such that the series ˆ H is asymptotic for Hj in Sj × D as z → 0, h(z) − z = o(zk+1).

The map TO for complex saddlenodes

Theorem The correspondence mapping ∆: (R+, 0) → (R+, 0) TO a center manifold of a degenerate elementary singular point of a real-analytic vector field is the restriction to (R+, 0) of any of the two compositions ∆ = g u ◦ ∆st ◦ F u = g l ◦ ∆st ◦ F l, where F is the normalizing map-cochain (see the theorem on sectorial normalization) for the corresponding monodromy transformation F u (F l) is that mapping in the tuple F that is defined in the sector Su (Sl) adjacent from above (respectively from below) to (R+, 0) and the germs g u, g l are holomorphic and have a positive derivative at their fixed point zero.

Lecture 5. Conditionary proof of the finiteness theorem for the alternant polycycles

Transition to the logarithmic chart Mapping in a natural chart The same mapping in the loga- rithmic chart 1 Power: z → Czν Affine: ζ → νζ − ln C 2 Standard flat: z → exp(−1/z) Exponential: ζ → exp ζ 3 A mapping defined in a sector with vertex 0 and expandable in a con- vergent

• r

asymptotic Taylor series = z(1 + ∞

1 ajzj)

A mapping defined in a horizon- tal half-strip and expandable in a convergent or asymptotic Du- lac (exponential) series = ζ + ∞

1 bj exp(−jζ)

Transition to the logarithmic chart (continued)

4 hk,a : z → kzk(1 − az−k ln z)−1 ˜ hk,a : ζ → kζ − ln k − ln(1 − aζ exp(−kζ)) 5 An almost regular map- ping with asymptotic Dulac series at zero z → Czν + Pj(z)zνj, where C > 0, ν > 0, < νj ր ∞, and and the Pj are real polynomials An almost regular mapping with asymptotic Dulac exponential se- ries at infinity ζ → νζ − ln C + Qj(ζ) · exp(−µjζ) where C > 0, ν > 0, 0 < µj ր ∞, and the Qj are real polynomials

Normalizing cochains in the logarithmic chart: cochains of class NCk

Upon transition to the logarithmic chart the normalizing cochain F becomes a map-cochain defined in a half-plane C+

a : ξ ≥ a; a depends on the cochain.

• 1. The k-partition of a punctured disk by sectors becomes the

partition of C+

a into half-strips by the rays η = πm/k, m ∈ Z.

• 2. The components of the map-cochain extend analytically to

the ε-neighborhoods of the corresponding half-strips in the partition for arbitrary ε ∈ (0, π/2) (a depends also on ε).

• 3. They have an exponentially decreasing correction

(difference with the identity).

• 4. The modulus of the coboundary has the upper estimate

exp(−C exp kξ) for some C > 0 depending on the cochain.

• 5. The mappings making up F can be expanded in a common

asymptotic Dulac exponential series; see row 5 of the table.

Final structural theorem

Theorem (final structural theorem) The monodromy map of an alternant polycycle belongs to the group: ∆ ∈ Group(AH0, J0, Aff ) ∩ MR where the groups J0 and H0 are defined below, MR is the set

• f maps that are real on (R+, ∞).

Preliminary structural theorem

Theorem (Preliminary structural theorem) The monodromy map of an alternant polycycle belongs to the group G1 =+ Gr(R, FROM ◦ R ◦ TO), whose R is the group of almost regular germs, TO is the set

• f Dulac maps TO the center manifold of a real analytic

saddlenode, and FROM = TO−1. We will deduce the final structural theorem from the preliminary one.

Cochains of class NC

Let ak = kζ − ln k NC = ∪Ad(a−1

k )NCk.

All the cochains in this class written in the logarithmic chart correspond to one and the same partition of the half-plane C+

a : ℜζ > a (a depends on the cochain) by the rays η = πl.

The partition by these rays of an arbitrary domain in the right half-plane is denoted by Ξst and is called the standard partition. The domains of this partition are the half-strips Πj : η ∈ π(j − 1), πj, ξ ≥ a. The half-strips Π0 and Π1 adjacent to (R+, ∞) are called the main ones.

Cochains of class NC

• 1. Each cochain F ∈ NC corresponds to the standard

partition of some right half-plane C+

a , where a depends on F.

• 2. All the mappings in the cochain extend to the

ε-neighborhoods of the corresponding half-strips for some ε > 0. The mappings in the cochain corresponding to the main half-strips Π0 and Π1 of the partition extend holomorphically to a germ at infiniy of a half-strip Π(ε)

∗

for any ε > 0. Here Π∗ = Φ0Π, Π = {|ℑζ| < π 2 , ℜζ > a} Π(ε)

∗

= ΦεΠ Φε = ζ + (1 − ε)z−2, ε ∈ [0, 1).

• 3. The corrections of all mappings of the cochain extended as

in the previous item may be estimated in modulus from above by the decreasing exponential exp(−µξ) for some µ > 0 common for all the mappings in F.

• 4. The correction of the coboundary δF in the

ε-neighborhoods of all the rays of the partition can be estimated from above by an iterated exponential: |δF − id| < exp(−C exp ξ) for some C > 0.

• 5. The cochain F may be expanded in an asymptotic Dulac

series in its domain, including the extended components mentioned in item 2.

Axioms

Cochains of class FC0 are defined in so called quadratic standard domains and those of class FC1 are defined in so called standard domains of class 1 defined below. Example: NC ⊂ FC0. Definition Denote by FCm

+ (m = 0, 1) the set of exponentially decreasing

cochains: F ∈ FCm

+ (m = 0, 1) ⇔ ∃ε > 0 : |F| < e−εξ in

some standard domain. Recall ∆ ∈ Group(AH0, J0, Aff ) ∩ MR

Axioms (continued)

Inclusion axiom J0 = Ad(Aff )A0 ⊂ Group(id + FC 0

+)

AH0 ⊂ Group(id + FC 1

+ ◦ exp ◦µ|µ > 0).

Here µ stands for the multiplication map by µ, µ : ζ → µζ, F ◦ exp ◦µ : ζ → F ◦ exp µζ. The first statement will be repeated later as part of Shift lemma 4.

Axioms (continued)

Theorem (Phragm´ en–Lindel¨

• f)

A simple or sectorial cochain that decreases on (R+, ∞) faster than any exponent is identically zero near (R+, ∞). In other words, let F ∈ FC0,1 and |F| ≺ exp(−νξ) ∀ν > 0 on (R+, ∞). Then F u ≡ F l ≡ 0. Theorem (LET — Lower estimate theorem) Let F ∈ FC0,1, and F ≡ 0. Then there exists ν > 0, such that |F| ≻ exp(−νξ) on (R+, ∞).

Axioms (continued)

The following shift lemmas are also taken as axioms. Lemma (SL1) Translation and shift by R0 preserves the class: F1

(+)µ ◦ (C + R0) = F1 (+)µ.

(1) FC 0 ◦ R = FC 0. (2)

Axioms (continued)

Lemma (SL2) a) Let ϕ ∈ FC 0

+ or ϕ ∈ F1 +µ and ψ ∈ F1 +ν, 0 < µ < ν. Then

ϕ ◦ (id + ψ) = ϕ + ˜ ψ, where ˜ ψ ∈ F1

+ν.

b) Let ϕ ∈ FC 0

+ or ϕ ∈ F1 +µ. Then (id + ϕ)−1 = id − ˜

ϕ, where ˜ ϕ ∈ FC 0

+ or ˜

ϕ ∈ F1

+µ respectively.

c) FC 0

+ ◦ (id + FC 0 +) = FC 0 .

In other words, statements b) and c) claim that cochains of the form id + FC 0

+, as well as cochains id + F1 +µ, form a

group under composition.

Axioms (continued)

Lemma (SL3) Let ϕ ∈ F1

+ν and ψ ∈ F1 +µ, 0 < µ < ν. Then

ϕ ◦ (id + ψ) ∈ F1

+ν.

The parameters ν, µ show in particular the rate of double-exponential decrease of the cochain components. Lemma (SL4) a)Ad(Aff )A0 ⊂ Group(id + FC 0

+)

b)F1

+µ ◦ (id + FC 0 +) = F1 +µ.

MDT and ADT: multiplicative and additive decomposition theorems

Theorem (Multiplicative decomposition theorem, MDT) The monodromy map ∆ can be decomposed into the following product: ∆ = a ◦ (id + ϕ) ◦

N

• j=1

(id + ψj), where ϕ ∈ FC 0

+, ψ ∈ F1 +µj, F1 +µj = FC 1 + ◦ exp ◦µj and µj are

positive and increase with j.

MDT and ADT: multiplicative and additive decomposition theorems

Theorem (Additive decomposition theorem, ADT) The monodromy map ∆ can be presented in the following form: ∆ = a + ˜ ϕ +

N

• j=1

˜ ψj, (3) where ˜ ϕ ∈ FC 0

+, ˜

ψj ∈ F1

+µj,.

Finiteness theorem for alternant polycycles may be easily derived from the ADT and LET.

Lecture 6. Model for the axioms: simple and sectorial cochains.

Standard domains of class 1

Let us now pass to the definitions of simple and sectorial

• cochains. These definitions have two parts: regularity and
• decomposability. Let us start with the regularity part.

Definition A standard domain of class 1 depending on two positive constants ε ∈ (0, 1) and C is Ωε,C = Ψε(C+

C),

where C+

C = C+\KC, KC = {|ζ| < C}, Ψε : KC → C+, ζ → ζ+ζ1−ε.

Simple partitions

Definition A standard partition of a domain Ω ⊂ C is a partition of Ω by horizontal lines Im ζ = πj, j ∈ Z. A domain Ω R+ convex, if it contains a ray a + R+ together with any point a. The domains of the standard partition of an R+ convex domain Ω are horizontal half strips of width π. Definition Let ρ be a biholomorphic map of a domain ˜ Ω into an R+-convex domain Ω. A partition of the type σ∗Σst, where σ = ρ−1 is a pull back of the standard partition under the map ρ. Namely, a domain of the partition σ∗Σst is the image of a domain Π ∩ ρ˜ Ω under the map σ, where Π is a domain of the standard partition of Ω.

Definition A simple partition of type σ = (µ1, . . . , µN), µ1 > µ2 · · · > µN > 0 (1) is a product of images of the standard partition: Σσ =

N

• 1

µj∗Σst. (2) Definition A generalized ε-neighborhood of a boundary curve γ of a partition σ (this curve is in fact a horizontal ray) is the union

• f the images µj(γε

j ) of the ε-neighborhoods of the boundary

rays γj of a standard partition over those j for which µjγj ⊃ γ.

Regular simple cochains

Definition Consider a standard domain and its simple partition of type (2). A rigging cochain mε,C,C ′ is defined in the generalized ε-neighborhood of the boundary ∂Σσ, and in a generalized ε-neighborhood of a boundary curve γ, it takes the form: mε,C,C ′ =

• C exp(−C ′ exp µ−1

j ξ).

(3) We now come to one of the main definitions of this lecture.

Definition A regular simple cochain F of type σ (2) in an R+-convex domain Ω is a cochain in Ω with the following properties:

• Partition. The cochain F corresponds to the partition Σσ.
• Extension. Components of F may be holomorphically extended

to the generalized ε-neighborhoods of the boundary rays of the corresponding domains of the partition Ξσ.

• Growth. All the components of F grow no faster than some

exponent : |F| ≺ exp νξ in Ω for some ν ∈ R. If ν may be chosen negative then F is rapidly decreasing.

• Coboundary. The coboundary of F admits an upper estimate

by some rigging cochain (3): ∃ ε, C, C ′ such that |δF| < mε,C,C ′ in the generalized ε-neighborhoods of ∂Ξσ.

Realizations

Definition Let σ be the same as in (2). A partition of a R+-convex domain of type (σ, k) is a simple partition of the type (σ, k) = (µk+1, . . . , µN). (4) We now need some special domains and their neighborhoods. Recall that the curvilinear strip Π∗ is Π∗ = ψ0Π, Π = {ξ ≥ a, |η| ≤ π 2 }, ψ0 : ζ → ζ + ζ−2. (5)

A generalized ε-neighborhood

A generalized ε-neighborhood of Π∗, Π(ε)

∗

is defined as Πε

∗ = ψεΠ, ψε : ζ → ζ + (1 − ε)ζ−2, for arbitrary ε ∈ (0, 1).

Let Πj = {ξ ≥ a, η ∈ [π(j − 1), πj]}. Definition Let σ be a partition (2). A domain Ωσ,k (domain of type (σ, k)) is Ωσ,k = µk(Π1 ∪ Π∗). (6) Its generalized ε-neighborhood is Ω(ε)

σ,k = µk(Πε 1 ∪ Π(ε) ∗ ).

Here Πε

1 is a usual ε-neighborhood of Π1.

Definition Let F be a simple cochain in a standard domain Ω that corresponds to a simple partition of the type (2). A simple cochain F(k) is called a k-realization of F provided that:

• Domain. It is defined in Ωσ,k
• Partition. It corresponds to a partition of type (σ, k), see (4).
• Extendability. The components of F(k) may be extended to the

generalized ε-neighborhoods of their domains.

• Coincidence. In the domain Ωσ,k ∩ Ω+ F(k) = F. Other

requirements on F(k) are contained in the definition of simple cochains. Example A cochain of class NC has the type σ = (µ1), µ1 = 1. Its 1-realization is the holomorphic function, the component F u of the cochain extended to Π∗. This extension is possible by the requirement 2 of the definition of the class NC.

Remark The coboundary of F(k) on (R+, ∞) is |∂F(k)|(R+,∞) ≺ C exp(−C ′ exp ◦µ−1

k+1)

whilst |∂F|(R+,∞) ≺ C exp(−C ′ exp ◦µ−1

1 ).

Therefore, when we replace F by F(k) we improve estimate of the coboundary on (R+, ∞), but reduce the domain where the cochain is defined. The realization are used in the proof of the Phragmen-Lindelof theorem for cochains.

Sectorial cochains

Definition A sectional partition of a standard domain Ω of the type exp σ = (exp µ1, . . . , exp µN), (7) where 1

2 > µ1 > · · · > µN > 0. is a partition of the form:

Ξexp σ = ΠN

1 (exp ◦µj)∗Ξst.

This is a partition of Ω by the rays arg ζ = πlµj provided that |πlµj| < π

2.

Note that the ray exp(R+ + iµ1π) does not intersect germs of standard domains at infinity for |µ1| = 1

2.

Definition A generalized ε-neighborhood of a ray L of a sectorial partition is a union ∪ exp µj(γε

j ) = L(ε); the summation is over

those j for which exp µjγj = L. Definition (rigging cochain of sectorial type) A rigging cochain ˜ mε,C,C ′ of sectorial type is defined in the generalized ε-neighborhood of the boundary rays of a sectorial

• partition. The component of ˜

mε,C,C ′ in the generalized ε-neighborhood L(ε) of the boundary ray L is ˜ mε,C,C ′|L(ε) =

• j

C exp(−C ′|ζ|µ−1

j );

(8) the summation is over those j for which L ⊂ (exp µj)Ξst.

Definition A sectorial cochain F of type (7) is a tuple of holomorphic functions with the following properties. Partition The cochain corresponding to sectorial partition Σexp σ of type (7). Extension Components of F may be holomorphically extended to the generalized ε-neighborhoods of the boundary rays of corresponding domains of the partition Σexp σ Growth All the components of F grow no faster than some exponent : |F| ≺ exp νξ in Ω for some ν ∈ R. If ν may be chosen negative then F is rapidly decreasing. Coboundary The coboundary of F admits an upper estimate by some rigging cochain (3): ∃ ε, C, C ′ such that |δF| < ˜ mε,C,C ′ in the generalized ε-neighborhoods of ∂Ξσ.

Realizations of sectorial cochains

These realizations are described in the same way as those for the simple cochains. This will be done in the next lectures. The class of the regular realizable simple (sectorial) cochains is denoted by FC0

reg (FC1 reg respectively).

This completes the regularity part of the construction. The decomposability is defined in Lecture 8. Lemma The spaces FC0

reg and FC1 reg are linear.

Lemma Let F ∈ FC0

• reg. Then there exists a realization F(k) of F such

that F(k) ◦ ln ∈ FC1

reg.

Classical Phragmen – Lindel¨

• f theorem

Theorem Suppose that D is a simply connected domain on the Riemann sphere that contains the point ∞ on its boundary. Assume that the function f is holomorphic in D and bounded and continuous on the closure of D, which is taken in the topology

• f C and does not contain ∞. Then

sup

D

|f | = sup

∂D

|f |. This theorem is a version of the maximum modulo principle for holomorphic functions.

Phragmen – Lindel¨

• f theorem for two quadrants

Theorem Suppose that the holomorphic function f : C+ → C+ is bounded on the union of the imaginary axis and the positive semi-axis and increases no more rapidly than an exponential in modulus: there exists a ν > 0 such that |f (ζ)| < exp ν|ζ|. Then f is bounded, and supC+ |f | = sup∂C+ |f |.

Corollary

Corrolary If a holomorphic function f : C+ → C is bounded and decreases on R+ faster than any exponential exp(−νξ), ν > 0, then f ≡ 0.

Cochains of class NC

• 1. Each cochain F ∈ NC corresponds to the standard

partition of some right half-plane C+

a , where a depends on F.

• 2. All the mappings in the cochain extend to the

ε-neighborhoods of the corresponding half-strips for some ε > 0. The mappings in the cochain corresponding to the main half-strips Π0 and Π1 of the partition extend holomorphically to a germ at infiniy of a half-strip Π(ε)

∗

for any ε > 0. Here Π∗ = Φ0Π, Π = {|ℑζ| < π 2 , ℜζ > a} Π(ε)

∗

= ΦεΠ Φε = ζ + (1 − ε)z−2, ε ∈ [0, 1).

• 3. The corrections of all mappings of the cochain extended as

in the previous item may be estimated in modulus from above by the decreasing exponential exp(−µξ) for some µ > 0 common for all the mappings in F.

• 4. The correction of the coboundary δF in the -neighborhoods
• f all the rays of the partition can be estimated from above by

an iterated exponential: |δF − id| < exp(−C exp ξ) for some C > 0.

• 5. The cochain F may be expanded in an asymptotic Dulac

series in its domain, including the extended components mentioned in item 2.

Phragmen – Lindel¨

• f theorem for the class NC

Theorem If a cochain F of class NC decreases on R+ faster than any exponential exp(−νξ), ν > 0, then F ≡ 0.

Trivialization of a cocycle

Lemma Let F be an ε -extendable cochain of class NC defined in an ε-neighborhood Ωε of a standard domain Ω. Let ΞF,ε (ΞF) be the partition of Ωε (respectively, Ω) that corresponds to this cochain . Let the coboundary of F be estimated from above by a m = exp(−C exp ξ), and m0 = sup

Ωε m,

• ∂ΞF,ε m ds = I < ∞.

Then there exists an ε–extendable functional cochain Φ defined in Ω, such that δF = δΦ

• n ∂ΞF,

max

Ω |Φ| ≤ Cε−1(m0 + I).

Maximum modulo principle for cochains

Lemma In assumptptions of the previous lemma, let Ω be a standard domain of class 1, F be a simple or sectorial or rotated sectorial cochain that grows no faster than the exponent exp νξ in Ω and is bounded on ∂Ω and on a positive real axis. Then F is bounded in Ω, and sup

Ω

|F| ≤ sup

∂Ω

|F| + 2Cε−1(m0 + I). where C, ε, m0, I are the same as in the previous lemma.

Preliminary estimate

Lemma Let F be a cochain of the class NC defined in a standard domain Ω , that decreases on R+ faster than any exponential. Then for any sector Sα: | arg ζ| < α < π/2 and any δ > 0 there exists C > 0 such that for any ζ ∈ Sα the following estimate holds: |F(ζ)| < exp(−C exp(1 − δ)ξ). The classical Phragmen – Lindel¨

• f theorem for a halfstripe

now implies the Phragmen – Lindel¨

• f theorem for the cochains
• f the class NC.

Simple partitions

Definition A standard partition of a domain Ω ⊂ C is a partition of Ω by horizontal lines Im ζ = πj, j ∈ Z. A domain Ω R+ convex, if it contains a ray a + R+ together with any point a. The domains of the standard partition of an R+ convex domain Ω are horizontal half strips of width π.

Definition A simple partition of type σ = (µ1, . . . , µN), µ1 > µ2 · · · > µN > 0 (1) is a product of images of the standard partition: Σσ =

N

• 1

µj∗Σst. (2) Definition A generalized ε-neighborhood of a boundary curve γ of a partition σ (this curve is in fact a horizontal ray) is the union

• f the images µj(γε

j ) of the ε-neighborhoods of the boundary

rays γj of a standard partition over those j for which µjγj ⊃ γ.

Regular simple cochains

Definition Consider a standard domain and its simple partition of type (1). A rigging cochain mε,C,C ′ is defined in the generalized ε-neighborhood of the boundary ∂Σσ, and in a generalized ε-neighborhood of a boundary curve γ, it takes the form: mε,C,C ′ =

• C exp(−C ′ exp µ−1

j ξ).

(3)

Definition A regular simple cochain F of type σ in an R+-convex domain Ω is a cochain in Ω with the following properties:

• Partition. The cochain F corresponds to the partition Σσ.
• Extension. Components of F may be holomorphically extended

to the generalized ε-neighborhoods of the boundary rays of the corresponding domains of the partition Ξσ.

• Growth. All the components of F grow no faster than some

exponent : |F| ≺ exp νξ in Ω for some ν ∈ R. If ν may be chosen negative then F is rapidly decreasing.

• Coboundary. The coboundary of F admits an upper estimate

by some rigging cochain : ∃ ε, C, C ′ such that |δF| < mε,C,C ′ in the generalized ε-neighborhoods of ∂Ξσ.

Realizations

Definition Let σ be the same as above. A partition of a R+-convex domain of type (σ, k) is a simple partition of the type (σ, k) = (µk+1, . . . , µN). (4) We now need some special domains and their neighborhoods. Recall that the curvilinear strip Π∗ is Π∗ = ψ0Π, Π = {ξ ≥ a, |η| ≤ π 2 }, ψ0 : ζ → ζ + ζ−2. (5)

A generalized ε-neighborhood

A generalized ε-neighborhood of Π∗, Π(ε)

∗

is defined as Πε

∗ = ψεΠ, ψε : ζ → ζ + (1 − ε)ζ−2, for arbitrary ε ∈ (0, 1).

Let Πj = {ξ ≥ a, η ∈ [π(j − 1), πj]}. Definition Let σ be a partition (2). A domain Ωσ,k (domain of type (σ, k)) is Ωσ,k = µk(Π1 ∪ Π∗). (6) Its generalized ε-neighborhood is Ω(ε)

σ,k = µk(Πε 1 ∪ Π(ε) ∗ ).

Here Πε

1 is a usual ε-neighborhood of Π1.

Definition Let F be a simple cochain in a standard domain Ω that corresponds to a simple partition of the type (2). A simple cochain F(k) is called a k-realization of F provided that:

• Domain. It is defined in Ωσ,k
• Partition. It corresponds to a partition of type (σ, k), see (4).
• Extendability. The components of F(k) may be extended to the

generalized ε-neighborhoods of their domains.

• Coincidence. In the domain Ωσ,k ∩ Ω+ F(k) = F. Other

requirements on F(k) are contained in the definition of simple cochains. Example A cochain of class NC has the type σ = (µ1), µ1 = 1. Its 1-realization is the holomorphic function, the component F u of the cochain extended to Π∗. This extension is possible by the requirement 2 of the definition of the class NC.

Remark The coboundary of F(k) on (R+, ∞) is |∂F(k)|(R+,∞) ≺ C exp(−C ′ exp ◦µ−1

k+1)

whilst |∂F|(R+,∞) ≺ C exp(−C ′ exp ◦µ−1

1 ).

Therefore, when we replace F by F(k) we improve estimate of the coboundary on (R+, ∞), but reduce the domain where the cochain is defined. The realization are used in the proof of the Phragmen – Lindel¨

• f theorem for cochains.

Sectorial cochains

Definition A sectional partition of a standard domain Ω of the type exp σ = (exp µ1, . . . , exp µN), (7) where 1

2 > µ1 > · · · > µN > 0. is a partition of the form:

Ξexp σ = ΠN

1 (exp ◦µj)∗Ξst.

This is a partition of Ω by the rays arg ζ = πlµj provided that |πlµj| < π

2.

Note that the ray exp(R+ + iµ1π) does not intersect germs of standard domains at infinity for |µ1| = 1

2.

Definition A generalized ε-neighborhood of a ray L of a sectorial partition is a union ∪ exp µj(γε

j ) = L(ε); the summation is over

those j for which exp µjγj = L. Definition (rigging cochain of sectorial type) A rigging cochain ˜ mε,C,C ′ of sectorial type is defined in the generalized ε-neighborhood of the boundary rays of a sectorial

• partition. The component of ˜

mε,C,C ′ in the generalized ε-neighborhood L(ε) of the boundary ray L is ˜ mε,C,C ′|L(ε) =

• j

C exp(−C ′|ζ|µ−1

j );

(8) the summation is over those j for which L ⊂ (exp µj)∂Ξst.

Definition A sectorial cochain F of type (7) is a tuple of holomorphic functions with the following properties. Partition The cochain corresponding to sectorial partition Σexp σ of type (7). Extension Components of F may be holomorphically extended to the generalized ε-neighborhoods of the boundary rays of corresponding domains of the partition Σexp σ Growth All the components of F grow no faster than some exponent : |F| ≺ exp νξ in Ω for some ν ∈ R. If ν may be chosen negative then F is rapidly decreasing. Coboundary The coboundary of F admits an upper estimate by some rigging (8): ∃ ε, C, C ′ such that |δF| < ˜ mε,C,C ′ in the generalized ε-neighborhoods of ∂Ξσ.

Realizations of sectorial cochains

These realizations are described in the same way as those for the simple cochains. This will be done in the next lectures. The class of the regular realizable simple (sectorial) cochains is denoted by FC0

reg (FC1 reg respectively).

This completes the definition of the classes FC0

reg and FC1 reg.

To define the classes FC0 and FC1 we have to add the decomposability properties.

Phragmen – Lindel¨

• f theorem for classes FC0

reg

and FC1

reg Theorem If a cochain of class FC0

reg or FC1 reg decreases on (R+, ∞)

faster than any exponent, then it is identically zero on (R+, ∞).