Horn maps of holomorphic functions locally pseudo-conjugate on their local parabolic basin Dimitri Le Meur

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Horn maps of holomorphic functions locally

pseudo-conjugate on their local parabolic basin

Dimitri Le Meur

Abstract

The lifted horn map hof a holomorphic function fwith a simple parabolic point at z0

is well known to be a local conjugacy complete invariant at z0; this is a classical result

proved independently by ´

Ecalle [´

E75], Voronin [Vor81], Martinet and Ramis [MR83].

Landford and Yampolski have shown in [LY] that, if two functions f1, f2with simple

parabolic points at z1, z2are globally conjugate on their immediate parabolic basins, with

the conjugacy and its inverse continuous at z1, resp. z2, their horn maps must be equal

as analytic ramiﬁed coverings. In this article, we introduce a notion of local conjugacy

on immediate parabolic basins, and show that two functions f1, f2with simple parabolic

points are locally conjugate on parabolic basins (without the hypothesis of continuity) if

and only if the non-lifted horn maps hiare equivalent as ramiﬁed coverings. This result

is a ﬁrst step to understand better invariant classes by parabolic renormalization.

arXiv:2210.11211v2 [math.DS] 25 Jan 2024

Contents

1 Introduction 3

2 Fatou coordinates, petals 8

3 Attracting, repelling cylinders and extension of Fatou coordinates 12

3.1 Attracting, repelling cylinders . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Every petal is brimming, uniqueness and expansion of Fatou coordinates . 14

3.3 Extension of Fatou coordinates . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Horn maps 20

5 Deﬁnition of local pseudo-conjugacy and immediate consequences 27

5.1 Intermediate statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Local semi-conjugacy on immediate basins . . . . . . . . . . . . . . . . . . 31

5.3 Local pseudo-conjugacies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Proof of the theorem 1.4 48

7 Parabolic point with several petals 61

7.1 Generalized deﬁnitions of semi-conjugacy, pseudo-conjugacy and reviewed

statements.................................... 61

7.2 A class of pseudo-conjugacies examples . . . . . . . . . . . . . . . . . . . . 63

References 65

1 Introduction

Let fbe a holomorphic function with a parabolic point at 0of multiplier 1and one

attracting axis. This is amounts to the fact that fadmits a Taylor expansion at 0of

the form f(z) = z+az2+o(z2)where a∈C∗. The pair of germs1

(h+

g, h−

(here the index gdenotes the initial of the word germ) at ±i∞of the lifted horn map

hof fis well known to be a local conjugacy complete invariant; this is a classical result

proved independently by ´

Ecalle [´

E75], Voronin [Vor81], Martinet and Ramis [MR83].

Theorem 1.1. Let f1,f2be holomorphic functions from an open neighborhood of 0∈C

to C, with the following Taylor expansion at 0:

fi(z) = z+aiz2+o(z2)

where ai̸= 0.

Suppose that f1,f2are conjugate in a neighborhood of 0via a local biholomorphism

ϕ. Then the germs at ±i∞,h1±

g,h2±

gof the horn maps h1,h2are equivalent as analytic

ramiﬁed coverings2via translations of Cat the domain and at the range. More precisely,

there exists σ, σ′∈C(independent of the sign ±) such that:

h1±

g=Tσ◦h2±

g◦Tσ′

where Tσdenotes the translation of the cylinder C/Zgiven by the formula z→z+σ.

Conversely, if there exists σ, σ′∈C(independent of the sign ±) such that

h1±

g=Tσ◦h2±

g◦Tσ′

then f1, f2are analytically conjugate in a neighborhood of 0.

The aim of this paper is to build up a complete invariant with a more ﬂexible covering

equivalence. We will introduce a notion of local pseudo-conjugacy designed for immediate

parabolic basins. We no longer work with germs of horn maps, but with bigger domains

of deﬁnition. Let D±⊂C/Zbe the connected components of the domain of the horn map

hcontaining a punctured neighborhood of ±i∞. Let h±be the map h:D±→h(D±).

For each open neighborhood U1of 0∈C, deﬁne U0

1to be the connected component

of U1∩Bf1

0containing a germ of the attracting axis off1, where Bf1

0is the immediate

parabolic basin of f1.

1Let z∈S, S′where S, S′are Riemann surfaces, and Uan open subset of Scontaining z. The germ

of the holomorphic function f∈ O(U, S′)(holomorphic function from Uto S′) at zis given by the

equivalence class of ffor the equivalence relation ∼over O(U, S′)deﬁned by g∼hif and only if there

exists an open set Vcontaining zsuch that g|V=h|V.

2This theorem is usually formulated in this way: h1and h2are equal modulo post-composition and

pre-composition by translations. We use there this terminology of analytic ramiﬁed covering equivalence

because it allows an analogy with the theorem 1.4 given below.

Deﬁnition 1.2. One says that ϕsemi-conjugates f1to f2on their immediate basins if

there exists U1an open neighborood of 0∈Csuch that ϕ:U0

1→Bf2

0is a semi-conjugacy,

namely:

ϕ◦f1(z) = f2◦ϕ(z)for z∈U0

1such that f1(z)∈U0

It is important to note that the range of ϕis not required to be a small neighborood

of 0, i.e. ϕis allowed to map points close to 0anywhere in the immediate basin of f2.

A semi-conjugacy ϕinduces a map ϕ:Bf1/f1→Bf2/f2in the following way: for

z=zmod f1∈Bf1/f1, there exists N∈Nsuch that for all n≥N,fn

1(z)∈U0

1. The

element ϕ(fn

1(z)) mod f2∈Bf2

0/f2is independent of the representative zof zand of the

chosen integer n≥N. We will see that ϕis necessarily a biholomorphism in proposition

5.17

Deﬁnition 1.3. Let (ϕ, ϕ′)denote a pair of local semi-conjugacies, i.e. ϕ:U0

1→Bf2

local semi-conjugacy of f1, f2and ϕ′:U0

2→Bf1

0local semi-conjugacy of f2, f1. We say

that (ϕ, ϕ′)is a local pseudo-conjugacy of f1, f2if ϕ:Bf1/f1→Bf2/f2has for inverse

ϕ′:Bf2/f2→Bf1/f1.

For two maps f1, f2to be locally pseudo-conjugate at 0, it is suﬃcient that they are

globally conjugate on their respective parabolic immediate basins.

Here is the principal theorem of the paper. It is proven in part 6in two separate

propositions (4propositions if one counts the complements):

Theorem 1.4. Let f1,f2denote holomorphic maps from an open neighborhood of 0∈C

to C, with Taylor expansion at 0:fi(z) = z+aiz2+o(z2)where ai̸= 0.

Suppose f1, f2locally semi-conjugate at 0on Bfi

0. Then there exists σ∈C/Zand a

pair of holomorphic maps ψ= (ψ+, ψ−), where ψ±:D±

1→ D±

2, such that:

h±

1=Tσ◦h±

2◦ψ±

and such that ψ±have removable singularities at ±i∞.

Conversely, if there exists σ∈C/Zand a pair of holomorphic maps ψ= (ψ+, ψ−),

where ψ±:D±

1→ D±

2, with removable singularities at ±i∞such that:

h±

1=Tσ◦h±

2◦ψ±

then f1, f2are locally semi-conjugate at 0on their immediate basins.

Furthermore, the maps ψ±have at ±i∞an expansion of the form ψ±(w) = w+ρ±+

o(1).

Remark 1.5. Theorem 1.1 involves the germs h±

1g, h±

2gof the lifted horn maps h1, h2

deﬁned on the subsets e

Di=π−1(Di)of C, where π:C→C/Zis the canonical projection,

whereas theorem 1.4 involves the maps h±

1,h±

2deﬁned on the subsets D±

iof the cylinder

C/Z.

Here is a complement of theorem 1.4:

Proposition 1.6. The theorem is still valid by replacing each occurrence of “semi-conjugacy“

by “pseudo-conjugacy“, and each occurence of “ψ= (ψ+, ψ−)pair of holomorphic maps

ψ±:D±

1→ D±

2“ by “ψ= (ψ+, ψ−)pair of biholomorphisms ψ±:D±

1→ D±

2“.

In this case, h±

1,h±

2are equivalent as analytic ramiﬁed covering via biholomorphism

between D±

1and D±

2(more precisely via ψ±), and a translation of C/Z(more precisely

via Tσ).

More explicitly. Let f1,f2denote holomorphic maps from an open neighborhood of

0∈Cto C, with Taylor expansion at 0:fi(z) = z+aiz2+o(z2)where ai̸= 0.

Suppose f1, f2locally pseudo-conjugate at 0on Bfi

0. Then there exists σ∈Cand a

pair of biholomorphisms ψ= (ψ+, ψ−), where ψ±:D±

1→ D±

2, such that:

h±

1=Tσ◦h±

2◦ψ±

and such that the maps ψ±have removable singularities at ±i∞.

Conversely, if there exists σ∈Cand a pair of biholomorphisms ψ= (ψ+, ψ−), where

ψ±:D±

1→ D±

2, with removable singularities at ±i∞such that:

h±

1=Tσ◦h±

2◦ψ±

then f1, f2are locally pseudo-conjugate at 0on their immediate basins.

Furthermore, the maps ψ±have at ±i∞an expansion of the form ψ±(w) = w+ρ±+

o(1).

Remark 1.7. Let us make the following remark whose demonstration we will omit. If we

replaced in theorem 1.4 the maps (h+,h−)by the germs (h+

g,h−

g), we would obtain a

statement that is a logical equivalence between the two following statements, which are

easily seen to be always true. The germs are equivalent as analytic ramiﬁed coverings

(this is true since the germs are inversible) if and only if f1, f2conjugate on the very

large attracting petals3included in a small neighborhood of 0(this is true since f1, f2

are conjugate to translations). Whence the interest to work with (h+,h−)for the study

of this more ﬂexible covering equivalence.

The book of Landford and Yampolsky [LY] proves the implication of theorem 1.4

with some supplementary hypotheses. Namely: ϕis a global biholomorphism on the

immediate basins Bf1

0,Bf2

0, and the conjugacy ϕand its inverse ϕ′=ϕ−1are continuous

at 0. The construction of ψwhich will be exposed here becomes, without these continuity

hypotheses, more diﬃcult.

It is expected that the theorem 1.4 may be naturally adapted for maps f1, f2with

parabolic points with any number (possibly distinct for f1, f2) of petals and cycles of

petals and for each of their horn maps. This conjectural generalisation of the theorem

will be explicitly formulated in part 7.

Theorem 1.1, up to renormalizing the Fatou coordinates, gives an equality between

h1±

g,h2±

g. This enables to study the dynamic of h±

gindependently of the choice of the

representative of fmodulo local conjugacy. The theorem 1.4, up to renormalizing the

3See deﬁnition 2.2

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Hornmapsofholomorphicfunctionslocallypseudo-conjugateontheirlocalparabolicbasinDimitriLeMeurAbstractTheliftedhornmaphofaholomorphicfunctionfwithasimpleparabolicpointatz0iswellknowntobealocalconjugacycompleteinvariantatz0;thisisaclassicalresultprovedindependentlyby´Ecalle[´E75],Voronin[Vor81],Martine...

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Horn maps of holomorphic functions locally pseudo-conjugate on their local parabolic basin Dimitri Le Meur

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