Small divisors in discrete local holomorphic dynamics

We give an introduction to the study of local dynamics of iterated holomorphic mappings near a ﬁxed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The ﬁrst part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to ﬁnd convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant ﬁxed points.


Introduction
In studying local dynamics of iterated holomorphic mappings near a fixed point, conjugations to simpler maps via formal power series can be constructed in a systematic manner. The question of convergence of these formal series is intimately related to the size of divisors occurring in their coefficients.
In these notes we give an account of the Poincaré-Dulac theory of simplifications by formal conjugation and the role of resonances, as well as classical small-divisor conditions and recent generalisations in one and several complex variables. We close with a recent application of a small divisor condition to understand non-trivial dynamics near a one-resonant fixed point in C 2 .

Local dynamics
In discrete local holomorphic dynamics, we study the behaviour of the iterates F •n , n ∈ N of holomorphic maps F : U → M from an open subsets U ⊆ M to a complex manifold M near a fixed point p = F( p) ∈ U . Clearly, arbitrary iterates are defined at p, but not necessarily on any fixed neighbourhood of p. The orbit of a point q ∈ U under F is the sequence {q n } n := {F •n (q) | n ∈ N, F •(n−1) (q) ∈ U }.
We say the orbit {q n } n is stable (in U ) if it is contained in U , i.e. {q n } n is infinite, and {q n } n escapes (from U ) if q n / ∈ U for some n ∈ N, i.e. {q n } n is finite. Let End(M, p) denote the space of germs of holomorphic self-maps of M fixing p. By abuse of notation, we use the same symbol for the germ F ∈ End(M, p) and a representative F : U → M.
Since we are only concerned with behaviour arbitrarily close to the fixed point p, we are really talking about the dynamics of germs, but we will sometimes pick representatives on arbitrarily small neighbourhoods U of p to talk about stable and escaping orbits.

Conjugation
A first step towards understanding the dynamics of a germ F ∈ End(M, p), is to conjugate F to a germ of a simpler form, such as a linear self-map of C d . We will be interested in holomorphic and formal conjugations.

Remark 2
We can immediately make some key observations: 1. For M = N this is just a change of local coordinates. 2. If F and G are conjugate, the local dynamics of F and G (as described in the previous section) are equivalent as conjugation preserves topological properties. 3. Taking a local chart ϕ : (M, p) → (C d , 0) centred at p, any holomorphic germ F ∈ End(M, p) is holomorphically conjugate to a germ G ∈ End(C d , 0), where d = dim M.
So for our purposes, it is enough to consider germs in End(C d , 0) and conjugations with elements of Aut(C d , 0), the set of invertible germs in End(C d , 0). Equivalently, these are (d-tuples of) convergent power series without constant term. Therefore, we can consider End(C d , 0) as a subspace of the space Pow(C d , 0) of (d-tuples of) formal power series with vanishing constant terms. In particular, we can conjugate via elements ∈ Pow(C d , 0) that are invertible with respect to formal composition of power series ( is invertible if and only if its linear part is). We also call an element ∈ Pow(C d , 0) a formal germ. Definition 3 Two formal germs F, G ∈ Pow(C d , 0) are formally conjugate, if there exists an invertible formal power series ∈ Pow(C d , 0) such that F = −1 • G • .
Clearly, holomorphic conjugacy implies formal conjugacy, but not vice versa. Note that formal conjugation does not necessarily preserve dynamical properties. However, formal conjugates are often easier to construct systematically. From there, one can try to prove convergence of the formal conjugating series. If this is not possible, one can always truncate the conjugating series and obtain: Lemma 4 Two germs F, G ∈ Pow(C d , 0) are formally conjugate if and only if they are polynomially conjugate up to every finite order, i.e. for every k ∈ N there exists an invertible polynomial P k ∈ Aut(C d , 0) such that for z ∈ C d near 0.
Hence, formal conjugacy ensures holomorphic conjugacy up to arbitrarily high orders. To derive dynamical properties of F ∈ End(C d , 0) from those of a formal conjugate G, we have to control the tail term O( z k ) in (1), for example via analytic estimates.

One complex variable
This section contains a survey of local formal and holomorphic conjugations in local dynamics in one complex variable and the related small-divisor problems. More details on local dynamics can be found in Milnor's book [34] and on small divisor problems in one complex variable in Marmi's lecture notes [33].
The local dynamics of a holomorphic map f ∈ End(C, 0) are determined at the first order by the derivative of f , which will be our first important invariant: Definition 5 Let f ∈ Pow(C, 0) be given by Then the number λ ∈ C is called the multiplier of f (at 0).

Formal conjugations
A major part of the formal classification is covered by the Poincaré-Dulac theorem on formal elimination of non-resonant terms (see also Definition 32): Proposition 6 (Poincaré [42], Dulac [14]) Let f ∈ Pow(C, 0) with multiplier λ and let Then f is formally conjugated to a power series g ∈ Pow(C, 0) with multiplier λ in normal form, that is, g contains only monomials z j for j ∈ Res(λ) or In the proof we will see that elimination of a monomial z j for j ∈ N in the expansion of f involves dividing by λ j − λ, hence for j ∈ Res(λ), we cannot in general eliminate the monomial z j by formal conjugation.
To prove the Proposition, we construct h and g solving (3) such that g is in normal form (2). With expansions f (z) = ∞ j=1 f j z j , g(z) = ∞ j=1 g j z j and h(z) = ∞ j=1 h j z j for f , g, and h, the homological equation (3) becomes Comparing coefficients of z j for j ∈ N, we have We now inductively choose the coefficients h j for j ∈ N such that . The values of g j for j ∈ Res(λ) follow from the construction. For j = 1, (5) is just f 1 h 1 = h 1 g 1 and we can freely choose h 1 ∈ C\{0}, so h is invertible and For j ≥ 2, all indices k, l 1 , . . . , l k in the sum on the right-hand-side of (5) are smaller than j, so the value of the sum is determined by the previous steps. Since f j is given, only the coefficients h j and g j are yet to be determined.
If j / ∈ Res(λ) or λ j − λ = 0, we can prescribe g j = 0 and solve uniquely for h j . If j ∈ Res(λ) or λ j − λ = 0, the left-hand-side of (5) vanishes, so h j is eliminated from the equation and g j is uniquely determined by the value of the sum on the right-hand-side. In particular, we can choose an arbitrary value for h j that will not influence g j in this step, but will be part of the sum on the right-hand-side in the later steps.
Thus, we have constructed h and g as desired. Moreover, up to arbitrary choices of h j for j ∈ Res(λ), the construction was uniquely determined by the prescribed normal form for g.

Remark 7
Proposition 6 and its proof give some immediate insights into the formal and holomorphic classification. In particular, the multiplier is a formal and holomorphic invariant.

Convergence in the hyperbolic case
If the multiplier λ of f ∈ End(C, 0) satisfies λ = 1, we call 0 a hyperbolic fixed point. For convenience, we also call the multiplier λ and the germ f hyperbolic in this case. The invertible hyperbolic case was the first for which unique holomorphic forms were found by Koenigs: Theorem 9 (Koenigs,[30]) Let f ∈ Aut(C, 0) be hyperbolic and invertible with multiplier λ. Then f is holomorphically conjugate to its linear part L λ : z → λz and the conjugation h ∈ Aut(C, 0) is uniquely determined up to multiplication by a non-zero constant.
Koenigs' proof was of dynamical nature. We show a proof based on majorant series first used by Siegel in [56], as this method will prove more versatile later.
Proof Since λ j = λ for all j ≥ 2, Proposition 6 provides a linearising series h ∈ Pow(C, 0) for f with coefficients solving (5). h is unique up to choosing h 1 = 0 and multiplying h 1 by μ = 0 multiplies each h j by μ. Assume without loss of generality, that h 1 = 1.
For all j ≥ 2, after a linear change of coordinates, we may assume f j ≤ 1 (by convergence of f near 0) and since λ = 1, there exists a c > 1 such that λ j − λ −1 < c for all j ≥ 2. To show convergence of h, we estimate the coefficients h j for j ≥ 2. By (5), we have Recursively applying the above estimate to any terms h l with l ∈ N, l ≥ 2 on the right-handside, this implies an estimate of h j by a sum where each term is a power of c multiplied by |h 1 | = 1. We separately estimate the number of summands and the powers of c in this expanded sum. The number of terms in the sum can be recursively expressed by σ 1 = 1 and Consider the generating series σ (t) := ∞ j=1 σ j t j and observe Solving for t and requiring σ (0) = 0 yields a unique holomorphic solution for small t, so σ converges near 0 and we have By induction on j ≥ 2, (6) implies and h converges.

Remark 10
The normal form L λ in the holomorphic category is unique by Remark 7. Hence invertible hyperbolic germs are uniquely determined up to holomorphic conjugation by their multiplier λ.

Parabolic normal forms and dynamics
A germ f ∈ End(C, 0) is called parabolic, if its multiplier λ at 0 is a root of unity.
If λ is a primitive p-th root of unity, then by Proposition 6, f is formally conjugate to the form with k ≥ 1 and a k = 0. This form is sufficient for studying the local dynamics, but far from a unique normal form. The formal classification is somewhat folklore, Voronin [59] and Arnol'd [4] attribute it to N. Venkov without a reference, whereas O'Farrel and Short [35] conjecture Kasner's articles [28,29] as the earliest source.
Proposition 12 (Formal classification) Let f ∈ End(C, 0) be parabolic with multiplier λ, a (primitive) p-th root of unity and f • p = id. Then there exist unique k ∈ N and b ∈ C such that f is formally conjugate to z → λz + z pk+1 + bz 2 pk+1 (10) (11) for small positively oriented loops γ around 0.
Proof We know f is formally conjugate to the form (9) and after a linear change of coordinates, we may assume a k = 1. A conjugation with (see (5)), so by suitable choice of c, we can ensure a k+ j = 0, if j = k. By induction on j ≥ 1, we can hence ensure a k+ j = 0 for all j ≥ 1, j = k and j ≤ j max up to any finite order j max ∈ N and by Lemma 4, we have formal conjugacy to the form (10). Equation (11) follows from the residue theorem and shows uniqueness of the parameter b.
This formal conjugation does not converge in general. In fact, the holomorphic classification of parabolic germs was achieved via infinite dimensional functional invariants by Écalle [16,17] and Voronin [59], showing no finite number of complex parameters can specify the holomorphic conjugacy class of a parabolic germ.
Nevertheless, the local dynamics admit a concise description: Let f ∈ End(C, 0) be tangent to the identity of order k + 1, i.e.
for z ∈ C near 0 and a = 0. After a linear change of coordinates, we may assume a = 1/k and Observe that the factor (1 − z k /k) will act contracting, whenever z k is a (small) negative real number, and expanding, whenever z k is a positive number, motivating the following definition: Definition 13 Let f ∈ End(C, 0) be of the form (12). Then a vector v ∈ C * is an attracting direction (for f at 0) if v k = −1 and a repelling direction, if v k = 1.
The attracting directions are precisely the k-th roots of unity v j = exp(2πi j/k) for j = 0, . . . , k − 1. A main tool in this setting is the coordinate w = ϕ(z) := z −k defined for z ∈ C * . For each j = 0, . . . , k − 1, the restriction of ϕ to the sector Let f : U → C be a representative of f . For a point z ∈ f (U )\{0} in the stable set and n ∈ N let z n := f •n (z) and w n := ϕ(z n ). Then for all n ∈ N (and z n near 0), we have From this, it is not hard to prove the celebrated Leau-Fatou parabolic flower theorem, proved in its initial form by Leau [31], and improved upon by Julia [27] and Fatou [19][20][21]: Theorem 14 (Parabolic flower theorem) Let f ∈ End(C, 0) be of the form (12). Then for every θ ∈ (0, π), there exists R θ > 0 such that for any R > R θ and j = 0, . . . , k − 1 the set is a parabolic petal for f , i.e. for z ∈ P j (R, θ), we have For f −1 the picture is rotated by π/k, so for θ > π/2, the parabolic petals for f and f −1 constructed above overlap near 0 and hence form a punctured neighbourhood U \{0} of 0. Hence the Parabolic flower theorem describes the local dynamics of f in a full neighbourhood of 0.

Small divisors in the elliptic case
A germ f ∈ End(C, 0) is called elliptic, if its multiplier is an irrational rotation λ = e 2πiθ with θ ∈ R\Q. This case can lead to the most complicated dynamical behaviour and has not been fully understood, even though formally any elliptic germ is linearisable by Corollary 8.
The question is when the (unique) formal linearisation converges, leads to small-divisor problems.
Recall that if f (z) = ∞ j=1 f j z j , (5) implies that a formal power series h ∈ Pow(C, 0) given by h(z) = ∞ j=1 h j z j with dh 0 = 1 linearising f is uniquely determined by for j ≥ 2. The series h diverges if the coefficients h j are too large too often. In light of the proof of Theorem 9, h may diverge only if the divisors λ j − λ are too small. This type of problem is known as a small divisor or small denominator problem and is closely related to the arithmetical properties of the rotation number θ . Pfeiffer first proved the existence of non-linearisable elliptic germs in [41]. Cremer continued Pfeiffer's work in [12] and [13] and proved the following result: if and only if lim sup

Remark 16 A set containing a countable intersection of dense open subsets of a topological
space is known as a comeagre or generic subset in the fields of topology and algebraic geometry. In particular, a generic set is necessarily dense and uncountable (by the Baire category theorem) and intersections with other generic sets are generic. Hence Theorem 15 shows that the set of multipliers admitting non-linearisable germs is dense and topologically generic in S 1 .
Proof Note that so (15) and (16) are finite or infinite at the same time. Let f (z) = ∞ j=1 f n z n with f 1 = λ = e 2πiθ and f j = e iθ j with where h j are given by h 1 = 1 and (14), so and lim sup n→∞ |h n | 1/n = +∞, hence the linearising series h diverges.
On the other hand, a few years later C. L. Siegel established guaranteed linearisability for many multipliers: Theorem 17 (Siegel [56]) If for λ ∈ C there exist C, μ > 0 such that for every m ≥ 2, then every germ f ∈ End(C, 0) with multiplier λ is holomorphically linearisable.
Siegel's proof introduced the majorant series method that we saw in the proof of Theorem 9. It is a special case of the proof of Brjuno's theorem.
In contrast to Remark 16, we will show in Lemma 22 that the set of multipliers for which all germs are holomorphically linearisable has full measure in S 1 . In fact Milnor [34,Lemma C.7] shows that the complement has Hausdorff dimension 0. So this set is large in the sense of measure theory, but topologically small as the complement of a generic set.

Lemma 19 λ = e 2πiθ satisfies the Siegel condition (17) if and only if θ is Diophantine.
Proof Let q ∈ N and p ∈ Z minimise |qθ − p|. By Pythagoras, we have

Remark 20
In other words, the Siegel condition (17) is equivalent to the rotation number θ being badly approximated by rationals.
be a polynomial of degree deg P = d, with P(θ ) = 0 and without rational zeros. Then for p/q ∈ Q, q > 0, we have q d P( p/q) ∈ Z\{0}, and hence This implies condition (18) with m = d. (17) is satisfied for λ in a subset S ⊆ S 1 of full Lebesguemeasure.

Lemma 22 The Siegel condition
Hence for the Lebesgue-measure μ, we have: The union U (k, c) := ∞ q=1 U (k, c, q) then has measure at most For k > 2 this sum is finite and for c 0 it converges to 0, so the set: has full measure: The strongest possible general linearisation result is due to A.D. Brjuno: Theorem 23 (Brjuno [9]) For λ ∈ C and an integer m ≥ 2, let for some/any strictly increasing sequence of positive integers 0 < p 0 < p 1 < · · · . Then every f ∈ End(C, 0) with multiplier λ is linearisable.
See the multi-dimensional version, Theorem 49 for the proof. (19) is independent of the chosen sequence { p j } j and is hence often represented as

Continued fractions and the Brjuno function
In this section, we give a brief overview of continued fractions as a powerful tool for more precise estimates on rational approximations and small-divisor problems in dimension 1. In higher dimensions, this tool is no longer available.
The continued fraction expansion of θ ∈ (0, 1)\Q is constructed inductively in the form of a sequence of positive integers {a j } j∈N and fractional parts {r j } j∈N ⊆ (0, 1] uniquely defined by r 0 = θ and the decomposition 1 r j−1 = a j + r j of 1/r j−1 into integer part a j and fractional part r j for j ≥ 1. The j-th convergent of the continued fraction expansion of θ is the number with p j , q j ∈ N relatively prime. Then p j /q j − −−→ j→∞ θ and we have: Hence the j-th convergent is the best rational approximation of θ with denominator at most q j+1 . Therefore, the convergents contain a lot of information about how well θ is approximated by rationals. For λ = e 2πiθ one can show the following: 1. Cremer's condition (15) is equivalent to: 2. Siegel's condition (17) is equivalent to: for every j ∈ N, for some C > 0 and β > 1.

The Brjuno condition (19) is equivalent to:
Cremer's condition (20) corresponds to fast growth of the denominators q j . That means after q j , there is a large gap until q j+1 q j meaning the error is small, or θ is well-approximated by rationals. In the same way, the Siegel and Brjuno conditions correspond to slow growth of denominators and can θ being badly approximated by rationals.
This point of view even allows an estimate of the radius of convergence of germs with a given multiplier: Let S λ denote the space of (germs of) univalent maps f : D → C with f (0) = 0 and f (0) = λ and for f ∈ S λ let h f denote the radius of convergence of a linearisation of f . For θ ∈ (0, 1), set Theorem 27 (Yoccoz [61] (see also [11])) There exists a constant C > 0, such that In the case of quadratic polynomials, we know that non-Brjuno multipliers imply small cycles (Theorem 25), which rule out linearisation. On the other hand, if we exclude small cycles, Brjuno's condition (21) can be weakened and still ensure linearisation: if and only if every f ∈ End(C, 0) with multiplier λ = e 2πiθ without small cycles is holomorphically linearisable. Otherwise, there exists f ∈ End(C, 0) with multiplier λ and a neighbourhood U ⊆ C of 0 such that every stable orbit of f in U has 0 as an accumulation point (in particular, f is not linearisable and does not have small cycles).

Remark 29
More on the so-called hedgehog dynamics near non-linearisable elliptic fixed points has been discovered by Pérez-Marco in [37,38] and Biswas in [5] relating the stable local dynamics to circle diffeomorphisms. In particular these techniques are strictly onedimensional and do not extend to neutral germs in several complex variables. Recently, Firsova, Lyubich, Radu, and Tanase [22,32] constructed hedgehogs in C d , d > 1 on invariant, real two-dimensional, C 1 -smooth centre manifolds by way of quasiconformal conjugation given attracting behaviour in the transversal directions, so all stable orbits accumulate on the centre manifold.

Several complex variables
In this section, we present the formal normalisations of Poincaré-Dulac in several complex variables and conditions for full or partial normalisations to converge. We are considering (formal) germs F ∈ Pow(C d , 0) with series expansion where P k is a d-tupel of homogeneous polynomials of degree k for each k ≥ 1. As before, the first object of interest is the linear part and, more specifically, its eigenvalues. (23). Then the eigenvalues λ 1 , . . . , λ d of the linear part d F 0 = P 1 (with repetition according to multiplicity) are called multipliers of F.
We may assume d F 0 to be in Jordan normal form. Like in one dimension, the dynamics of a germ F are described at the first order by its linear part.
Note The multipliers and Jordan block structure of d F 0 are invariants under holomorphic and formal conjugation.

Formal conjugation and resonances
In one complex variable, any non-parabolic invertible germ is formally linearisable, as the denominators λ − λ k never vanish in this case. To examine the situation in several variables, we introduce multi-index notation: . The construction of formal conjugations works analogously to the one variable case: Elimination of a monomial z α in the j-th component F j of F requires a denominator λ j −λ α . A major difference in several complex variables is that these can vanish even when none of the multipliers are parabolic depending on the arithmetic relations of λ j with the other multipliers. This phenomenon is known as resonance.

A resonance for the tuple
The order of the resonance (24) is |α|.
Note By the above definition, λ j = λ e j is not considered a resonance for F, while z j in the j-th component of F is considered a resonant monomial. This is so when we require the multipliers have no resonances, λ j = λ e j is not an obstruction, but we still consider z j a resonant monomial in the j-th component F j of a germ F.

Remark 33
On the other hand, λ j = λ k for j = k is a resonance for F, but we will not talk much about resonances of order 1, as the corresponding terms are predetermined by the Jordan normal form of d F 0 . This is a first example of the main difference to the one-dimensional case: even when none of the multipliers are zero or roots of unity, resonances can occur between different eigenvalues. Another example is λ = (1/2, 1/4) with the resonance (1/2) 2 = 1/4, which shows that there can be even purely attracting germs that cannot be formally linearised.
If we eliminate all terms without this obstruction, we arrive at a formal normal form: . . . , λ d ), and the expansion of G at 0 contains only resonant monomials, that is Then H conjugates F to G, if and only if H and G solve the homological equation or, comparing coefficients: for α ∈ N d \{0}, where e J := e j 1 + · · · + e j k . The sum on the right hand side contains only coefficients with index of order less than |α|. and the j-th component of the left hand side is with ε j ∈ {0, 1}. Hence, proceeding by induction in lexicographic order over (|α|, j), whenever λ α − λ j = 0, all coefficients of order less than |α| and h j α with j < j are given and we can choose h j α (uniquely) to ensure g j α = 0. Whenever λ α − λ j = 0, we can freely choose h j α . The resulting G has Poincaré-Dulac normal form by construction.

Remark 38
If F has resonances, the Poincaré-Dulac normal form is in general not unique. In particular, one Poincaré-Dulac normal form of F may be a divergent series, while another may be a convergent series or even a polynomial. In [2], Abate and Raissy describe an alternative approach to obtain unique formal normal forms.
In the formally linearisable case, we do have uniqueness (up to linear changes of coordinates):

Lemma 39 (Rüssmann [54]) If F ∈ End(C d , 0) is formally linearisable, then any Poincaré-Dulac normal form of F is linear.
This follows quickly from Theorem 35 and the observation that resonant terms commute with d F 0 in Jordan normal form.
As before, when we have a formal linearising series H for F, we would like to find conditions for that series to converge. A new problem is, that many classes of germs are not even formally linearisable. In those cases, we still find a formal conjugation H of F to a Poincaré-Dulac normal form G, but neither the conjugating series H , nor the normal form G are unique. A natural problem of interest is then to find conditions that ensure at least one of the normal forms G is a convergent series or even a polynomial and moreover that the conjugating series H converges.

Convergence in the Poincaré domain
We call F ∈ Pow(C d , 0) hyperbolic, if all its multipliers lie in C\S 1 . If F is invertible and all multipliers lie in D = { λ < 1} or in C\D = {|λ| > 1}, then we say F is in the Poincaré domain. Otherwise, we say F is in the Siegel domain.

) is a hyperbolic germ in the Poincaré domain that is formally linearisable (e.g. has no resonances), then F is holomorphically linearisable.
The proof is essentially the same as the one-dimensional case in Theorem 9 choosing h j α = 0, whenever we can. It is a special case of case of the reduced Brjuno linearisation Theorem 58 with much simpler estimates.
If we allow resonances, a simple, but crucial observation is the following: For a nice proof by Rosay and Rudin, see also the appendix of [53]. The finiteness of all Poincaré-Dulac normal forms and the numerical properties of the multipliers are both crucial parts of the proof. So in most other cases, holomorphic conjugacy to non-linear normal forms cannot be established in this way (see e.g. Theorem 55).
By Lemma 39, the above implies in particular:

Stable manifolds in the Siegel domain
If a hyperbolic germ F ∈ End(C d , 0) is in the Siegel domain, i.e. has multipliers inside and outside the unit disk, a Poincaré-Dulac normal form can be infinite. Even without resonances, to ensure convergence of the formal linearisation, we need additional assumptions, like the Brjuno condition (26). However, even without that, we still get a complete description of the dynamics by the stable manifold theorem, proved in C 1 by Perron [40] and Hadamard [23] (translated to English by Hasselblatt [24]) and in the holomorphic case by Wu [26] (see also Abate [1]). One can think of the stable and unstable manifolds as deformed coordinate planes. In the invertible case, these can be topologically "straightened out" yielding one possible proof of the Grobman-Hartman theorem (see e.g. Shub [55], Hasselblatt and Katok [25], or Abate [1]).

Brjuno's theorem
Outside the Poincaré domain, even formally linearisable germs may not admit a holomorphic linearisation. The convergence of formal linearisation depends on the denominators λ α − λ j for α ∈ N d and j ≤ d. The Brjuno condition (19) generalises to several variables as follows: Definition 47 For a tuple λ = (λ 1 , . . . , λ d ) ∈ C d and an integer m ≥ 2, let We say the numbers λ 1 , . . . , λ d satisfy the Brjuno condition, if for a strictly increasing sequence of positive integers { p i } i∈N .

Remark 48
We can, without loss of generality, use p k = 2 k in the above definition (and its generalisations below), but unlike in dimension one, there is no known number theoretic version of the Brjuno condition in terms of continued fractions. The Brjuno condition implies in particular, that there are no resonances.
Brjuno's linearisation theorem then looks just line in one variable and will be proved at the end of this section.

Remark 50
In more than one variable, it is not known whether the Brjuno condition (26) is strict, in that we do not know if for any tuple (λ 1 , . . . , λ d ) not satisfying the Brjuno condition, there exists a non-linearisable germ with those multipliers.
In some cases, we have negative results: If a single multiplier does not satisfy the onedimensional Brjuno condition, it is easy to construct a non-linearisable germ from a onedimensional one. A bit more interesting is the higher-dimensional analogue of Cremer's theorem 15, that we do not prove here:

be without resonances and such that
Then there exists a germ F ∈ End(C d , 0) with linear part diag(λ) that is not holomorphically linearisable.
Sternberg's theorem follows directly from Brjuno's theorem 49, that we prove now:
We can now show Brjuno's estimate on N j m (α): Proof We fix m and j and proceed by induction on |α|.
If 2 ≤ |α| ≤ m, we have for all 0 ≤ l ≤ s, so N j m (α) = 0. If |α| > m, we take the chosen decomposition (35) and note that only |β 1 | may be greater than K = max{|α| − m, m}. If |β 1 | > K , we decompose δ β 1 in the same way and repeat this at most m − 1 times to obtain a decomposition with 0 ≤ k ≤ m − 1, l ≥ 2 and In particular, (39) implies |α − α k | < m. Hence Lemma 53 shows that at most one of the εfactors in (38) can contribute to N j m (α) and we have To estimate the product (36) we partition the indices into sets (recall for I 0 the convention ω(1) = +∞). By Lemma 54, we have This bound is independent of α finite by the Brjuno condition (26). Hence with (33) and (34) it follows that and thus H converges.

Generalised Brjuno conditions
There have been several generalisations of Brjuno's condition to germs with resonances. The Theorems in this section emerge from variations of the proof of Brjuno's theorem 49.
A strong obstruction to convergence of normalising series are Jordan blocks: Theorem 55 (Yoccoz [61]) Let A ∈ C d×d be an invertible matrix in Jordan normal form with a non-trivial Jordan block associated to an eigenvalue of modulus 1. Then there exists a germ F ∈ End(C d , 0) with d F 0 = A that is not holomorphically linearisable.
If there are no resonances except the multiple eigenvalue of the Jordan block, then such a germ is formally linearisable, but not holomorphically linearisable. Hence from here on, most results will assume a diagonal linear part of F.

Remark 56
On the other hand, Écalle and Vallet constructed, without restrictions, holomorphic normal forms "nearby" the original germ in [18]. A deformation approach to holomorphic normalisation by Pérez-Marco was published in [39]. These will not be subject of these notes.
In 1977, Rüssmann stated a generalised version of Brjuno's condition allowing for resonances (later published in [54]). This condition is equivalent to the reduced Brjuno condition Raissy formulated in [48] (see [49,Thm. 4.1] for her proof of equivalence) and simply ignores the zero-divisors resulting from resonances:
Pöschel's partial linearisations extend to full linearisations, if F has a "nice" structure along the remaining directions, like a curve of fixed points (Rong [52]) or a so-called osculating manifold (Raissy [47]).
A partial linearisation as in Theorem 60 can be seen as the elimination an infinite subset of monomials from the series expansion of F such that the remaining non-vanishing coefficients do not interfere with elimination of monomials from the specified subset. This concept is expanded in the proof of Theorem 64, that generalises Pöschel's Theorem 60.
We first introduce the notion of a Brjuno set of exponents (as defined by the author in [51]):

Definition 61
Let F be a germ of endomorphisms of C d with where for k ≥ 2.

Remark 62
Now we can state a generalisation of Theorems 49 and 60 in the context of eliminating infinite families of monomials. The conditions are rather technical, but in the proof it becomes clear that they are just the necessary conditions to make this proof work. 1. If α ∈ A 0 and β ≤ α, then β ∈ A 0 , and if α ∈ A and β ≤ α, then β ∈ A 0 ∪ A. 2. If β 1 , . . . , β l ∈ A 0 and β 1 + · · · + β l ∈ A 0 ∪ A, |β 1 | ≥ 2 and f j 1 β 1 · · · f j l β l = 0, then e j 1 + · · · + e j l / ∈ A.

A is a Brjuno set for F.
Then there exists a local biholomorphism H ∈ Aut(C d , 0) conjugating F to G = H −1 •F •H where G(z) = |α|≥1 g α z α with g α = f α for α ∈ A 0 and g α = 0 for α ∈ A.
Remark 66 If we assume α∈A 0 f α z α to be in Poincaré-Dulac normal form, the condition f j 1 β 1 · · · f j l β l = 0 can be replaced by λ β m = λ j m for 1 ≤ m ≤ l to avoid dependence of Condition (2.) on the specific germ F.
Proof Assume by induction on k 0 , that f α = 0 for α ∈ A\A k 0 . We show that A 0 = A k 0 −1 and A = A k 0 satisfy the prerequisites of Theorem 64.
Therefore Theorem 64 shows that F is conjugate to G with g α = f α for α ∈ A 0 and g α = 0 for α ∈ A.
The proof of Theorem 64 emerges largely by careful examination of the first lines of the proof of Theorem 49: Comparing coefficients for α ∈ N d \{0}, this means

Proof of Theorem 64 Formal series
where e J := e j 1 + · · · + e j k . Take d H 0 = id and h α = 0 for α / ∈ A, |α| ≥ 2. Then for α ∈ A 0 , the first term in the sum vanishes by Condition (1.) and the second term vanishes by Condition (2.), so g α = f α . For α ∈ A, λ α id − is invertible by Condition (3.), the first term in the sum on the right hand side depends only on h-terms with index of order less than |α| and the second term vanishes again by Condition (2.). Hence (28) determines h α uniquely by recursion and we obtain a formal solution H such that G = H −1 • F • H has the required form.
Moreover, since the only non-vanishing coefficients h α occur for α ∈ A, we can estimate those h α in the same way as in the Proof of Brjuno's theorem 49, replacing ω by ω A and again using that the second term of the sum in (44) vanishes for these α. It follows that H converges and hence so does G.
forms even if they are not holomorphically conjugate. Our example is a so-called one-resonant germ (Bracci,Zaitsev [6]), but the techniques presented potentially extend to other neutral germs with non-trivial resonances, such as multi-resonant germs (Bracci,Raissy,Zaitsev [7]) or elliptic germs with a finite set of resonances.
We study a one-resonant germ F ∈ End(C 2 , 0) with a non-linear Poincaré-Dulac normal form G ∈ End(C 2 , 0) whose stable dynamics we understand. We then proceed in three steps establishing the local dynamics in increasing detail: 1. Finding a formal conjugation of F to a non-linear germ G in (finite) Poincaré Let F ∈ End(C 2 , 0) be of the form where λ = e 2πiθ , θ ∈ R\Q is an irrational rotation and l ∈ N, l > 5. We will later assume that λ satisfies the Brjuno condition (19). This is a so-called one-resonant germ, as defined and studied by Bracci and Zaitsev in [6], that is, the resonances of (λ, λ) = (λ, λ −1 ) are precisely the relations (λλ −1 ) k λ ±1 = λ ±1 , k ∈ N. Hence F has a Poincaré-Dulac normal form G such that with g 1/2 (u) = O(u l/2 ). In fact, Bracci and Zaitsev prove an analogue of Proposition 12 giving finite normal forms in the one-resonant case, which implies we can take g 1/2 ≡ 0 in (46). Since λλ = 1 by assumption, the projection π to a one dimensional variable u := π(z, w) := zw satisfies This defines a parabolic germ ∈ End(C, 0) with an attracting petal P 0 according to Theorem 14 and since •n (u) ∼ −n −1 , the factor (1 − zw 2 ) in (46) is contracting for u ∈ P 0 or (z, w) ∈ π −1 (P 0 ). The original germ F acts on the u-variable via To preserve the dynamical behaviour of in the u-variable, we need to bound the tail in terms of u, say in the set W (β) := {(z, w) ∈ C 2 | |z|, |w| < |u| 1/β } = {|z| β−1 < |w| < |z| 1/(β−1) } for β > 2. Bracci, Zaitsev, Raissy, and Stensønes [6,8] show that stable orbits of (z, w) ∈ W (β) near 0 are precisely those following the parabolic dynamics of . In particular, the proof of shows: Theorem 67 (Bracci,Zaitsev [6]) For F ∈ End(C 2 , 0) of the form (45) with l > 5 there exist β > 2 and a parabolic petal P 0 for as in Theorem 14, such that the open set is a uniform local basin of attraction for F, that is F(B) ⊆ B, and lim n→∞ F •n ≡ 0 uniformly in B.
In the same way −P 0 is a repelling petal for and −B is a uniform local basin of attraction for F −1 .

Remark 68
The local basin B has some notable properties: It is doubly connected and invariant under (z, w) → (e it z, e −it w) for t ∈ R by definition, meaning each (z, w) ∈ B is part of a curve "winding" around the origin in B. Moreover, B avoids the coordinate axes, but contains line-segments from the origin in any other complex direction.
Unlike the parabolic case in one variable in Theorem 14, the local basins B and −B do not form a full neighbourhood of the origin, so the local dynamics are not completely determined (unless F is in Poincaré-Dulac normal form, see Bracci,Raissy,and Zaitsev [7,Thm. 5.3]).
In particular, there could still be more attracting orbits that never end up in B, possibly comprising a connected locally uniform basin of attraction containing B not contained in the iterated preimage n∈N F •(−n) (B).
To exclude this possibility, Bracci, Raissy, and Stensønes [8] assume in addition that λ satisfies the Brjuno condition (19). Given the resonances generated by λλ = 1, some algebraic manipulation shows that this is already equivalent to the partial Brjuno condition (41) for λ and λ in (λ, λ), so Pöschel's partial linearisation theorem 60 provides a Siegel disk through the origin tangent to each axis, that is an F-invariant disk on which F is conjugate to an irrational rotation. In other words, F is holomorphically conjugate to It then follows that the local basin B and also any larger basin of attraction cannot intersect these rotating disks, hence we can restrict to orbits in D * × D * . Since W (β 0 ) covers all directions apart from the axes and inside W (β 0 ) the dynamics are controlled by the parabolic dynamics of , an orbit converging to the origin outside B has to converge tangentially to the axes. On the other hand, the directions of an orbit in B stay away from the axes. Thus the Kobayashi distance with respect to D * × D * between two such orbits grows large as they approach 0, contradicting the non-expansiveness of the Kobayashi distance. This shows: Theorem 69 (Bracci, Raissy, Stensønes [8]) Let F ∈ End(C 2 , 0) be of the form (45), λ a Brjuno number, l ∈ N and B as in Theorem 67. Then the component of n∈N F •(−n) (B) containing B is a maximal connected uniform basin of attraction at 0.
We want to understand not just the maximal basin for B, but also any possible other basins at 0. Returning to the form (48) of F, we see that the tail is of order O(u) near 0. This is not small enough to prevent it from interfering with the parabolic dynamics of u. We want to eliminate more terms using Theorem 65. Further algebraic manipulation using the relation λλ = 1, shows that the Brjuno condition (19) on λ is equivalent to A k = {β ∈ N 2 | |β| ≥ l, min{β 1 , β 2 } = k − 1} being a Brjuno set (42) for F for every k ∈ N, k ≥ 1. Let A 0 = {|β| < l}. To verify the rather technical Condition (2.) of Theorem 65, the key observation is that in the germ (45) for β ∈ A 0 with f j β = 0, we have β ≥ e j . This follows from the fact that F is in Poincaré-Dulac normal form up to any orders contained in A 0 and resonances of F in the j-th component have exponent β = m · (1, 1) + e j for some m ∈ N. Hence Theorem 65 applies to A 0 , . . . , A k 0 for any k 0 ∈ N. Observing that A k corresponds to terms in O(u k−1 ), but not O(u k ), the elimination of those terms lets us holomorphically conjugate F to with u = zw and some 3 ≤ k 0 ≤ l/2. Hence, on a small enough neighbourhood U ⊆ C 2 of 0, the tail is of order O(u 3 ) and small enough to not interfere with the parabolic dynamics of in the u-variable. Now near 0 not only the stable orbits in W (β 0 ), but any stable orbits outside the Siegel disks will follow the parabolic dynamics of the u-variable and we can improve Theorem 69 to: Theorem 70 (R. [51]) Let F ∈ End(C 2 , 0) be of the form (45), λ a Brjuno number, l ∈ N and B as in Theorem 67. Then F admits a Siegel disk tangent to each axis and for small neighbourhoods U ⊆ C 2 of 0, any stable orbit in U is either contained in one of the Siegel disks or eventually contained in the local basin B.
Applying the theorem to the inverse F −1 , we can understand the local dynamics of all points with stable forward or backward dynamics:

Conflict of interest
The corresponding author states that there is no conflict of interest.
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