Periodic cycles of attracting Fatou components of type $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ in automorphisms of $\mathbb{C}^{d}$

TL;DR

This paper constructs automorphisms of complex spaces with multiple attracting Fatou components arranged in cycles, each biholomorphic to a product space, and analyzes their dynamical properties near a fixed point.

Contribution

It generalizes previous examples to create automorphisms with arbitrary finite cycles of Fatou components of a specific biholomorphic type, including invariant and periodic arrangements.

Findings

Automorphisms admit any finite number of non-recurrent Fatou components.

Each Fatou component is biholomorphic to ^{*})^{d-1} and attracts to a common boundary fixed point.

No orbit in these components can converge tangent to a complex submanifold.

Abstract

We generalise a recent example by F. Bracci, J. Raissy and B. Stens{\o}nes to construct automorphisms of $\mathbb{C}^{d}$ admitting an arbitrary finite number of non-recurrent Fatou components, each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ and all attracting to a common boundary fixed point. These automorphisms can be chosen such that each Fatou component is invariant or such that the components are grouped into periodic cycles of any common period. We further show that no orbit in these attracting Fatou components can converge tangent to a complex submanifold, and that every stable orbit near the fixed point is contained either in these attracting components or in one of $d$ invariant hypersurfaces tangent to each coordinate hyperplane on which the automorphism acts as an irrational rotation.

Equations

$$\mathcal{F}:=\{z\in {\mathbb{C}}^d\mid {\{F^n\}}_{n\in \mathbb{N}}\text{ is normal on a neighbourhood of }z\}.$$

$$F(z^{1},\ldots,z^{d})=(\lambda_{1}z^{1},\ldots,\lambda_{d}z^{d})\Big{\lparen}1-\frac{(z^{1}\cdots z^{d})^{k}}{kd}\Big{\rparen}+O(\lVert z\rVert^{l}),$$

$$({\zeta }^1,\cdots ,{\zeta }^d)\mapsto ({\zeta }^1+1,{\zeta }^2,\dots ,{\zeta }^d).$$

$$F_{\mathrm{N}}\lparen z\rparen=\Lambda z\cdot\Big{\lparen}1-\frac{(z^{1}\cdots z^{d})^{k}}{kd}\Big{\rparen},$$

$$S_{h}(R,\theta):=\big{\{}u\in\mathbb{C}\mid{\big{\lvert}u^{k}-\tfrac{1}{2R}\big{\rvert}}<\tfrac{1}{2R},{\big{\lvert}\arg(u)-\tfrac{2\pi h}{k}\big{\rvert}}<\theta\big{\}}$$

$$H(R,\theta ):=\{U\in \mathbb{C}\mid \mathrm{Re}U>R,|\arg (U)|<k\theta \}.$$

$$W(\beta ):=\{z\in {\mathbb{C}}^d\mid |z^j|<|\pi (z){|}^{\beta }\text{ for }j\le d\},$$

$$B_h(R,\theta ,\beta ):=\{z\in W(\beta )\mid \pi (z)\in S_h(R,\theta )\}.$$

$$F(z)=F_{\mathrm{N}}(z)+O(\Vert z{\Vert }^l),$$

$$\begin{gathered} \{(u,z^{\prime })\in ({\mathbb{C}}^{\ast }{)}^d\mid u\in S_h(R_0,{\theta }_0),|u{|}^{1-{\beta }_0}<|z^2\cdots z^d|,|z^j|<|u{|}^{{\beta }_0}\text{ for }j\ge 2\} \\ =\{u\in S_h(R_0,{\theta }_0),|u{|}^{1-(d-j+1){\beta }_0}|z^2\cdots z^{j-1}{|}^{-1}<|z^j|<|u{|}^{{\beta }_0}\text{ for }j\ge 2\}. \end{gathered}$$

$$B_h=\{(z,w)\in W(\beta )\mid |\arg (z)+\arg (w)-2\pi h/k|<\theta \}.$$

$$\{(r_1,r_2)\in {\mathbb{R}}_{+}^2\mid r_1^{\frac{1-\beta }{\beta }}<r_2<r_1^{\frac{\beta }{1-\beta }}\}\times \{(s,t)\in {\mathbb{T}}^2\mid d_{S^1}(s+t,2\pi h/k)<\theta \}\subseteq {\mathbb{R}}_{+}^2\times {\mathbb{T}}^2$$

$$d{F}_0=\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_d).$$

$$\sum _{k\ge 1}{2}^{-k}\log {\omega }_A^{-1}(2^k)<\infty ,$$

$${\omega }_A(k):=\min \{|{\lambda }^{\alpha }-{\lambda }_i|\mid \alpha \in A,2\le |\alpha |\le k,1\le i\le d\}\cup \{1\}$$

$$G(z)$$

$$({\lambda }^{\alpha }\mathrm{id}-\Lambda )h_{\alpha }=f_{\alpha }-g_{\alpha }+\sum _{2\le k<|\alpha |}\sum _{j_1\le \cdots \le j_k}\sum _{{\beta }_1+\cdots +{\beta }_k=\alpha }(f_{e_J}{h}_{{\beta }_1}^{j_1}\cdots h_{{\beta }_k}^{j_k}-h_{e_J}{g}_{{\beta }_1}^{j_1}\cdots g_{{\beta }_k}^{j_k}),$$

$$\underset{\alpha \in {\mathbb{N}}^d}{\sup }\frac{1}{|\alpha |}\log \Vert h_{\alpha }{\Vert }_1<\infty .$$

$$\Vert h_{\alpha }{\Vert }_1\le d\cdot \Vert h_{\alpha }{\Vert }_{\infty }\le d\cdot {\epsilon }_{\alpha }^{-1}\sum _{2\le k\le |\alpha |}\sum _{{\beta }_1+\cdots +{\beta }_k=\alpha }\Vert h_{{\beta }_1}{\Vert }_1\cdots \Vert h_{{\beta }_k}{\Vert }_1,$$

$${\sigma }_r:=d\sum _{k=2}^r\sum _{r_1+\cdots +r_k=r}{\sigma }_{r_1}\cdots {\sigma }_{r_{\nu }}\text{for }r\ge 2,$$

$${\delta }_{\alpha }$$

$$\Vert h_{\alpha }{\Vert }_1\le {\sigma }_{|\alpha |}{\delta }_{\alpha }$$

$$\sigma (t)$$

$$\displaystyle=t+d\sum_{k=2}^{\infty}\Big{\lparen}\sum_{r=1}^{\infty}\sigma_{r}t^{r}\Big{\rparen}^{k}$$

$$=t+d\frac{\sigma (t{)}^2}{1-\sigma (t)}.$$

$$\sigma (t)=\frac{1+t-\sqrt{(1+t{)}^2-4(d+1)t}}{2(d+1)}$$

$$\underset{r\ge 1}{\sup }\frac{1}{r}\log {\sigma }_r<\infty .$$

$${\delta }_{\alpha }={\epsilon }_{\alpha }^{-1}{\delta }_{{\beta }_1}\cdots {\delta }_{{\beta }_k}$$

$${\delta }_{\alpha }={\epsilon }_{{\alpha }_0}^{-1}{\epsilon }_{{\alpha }_1}^{-1}\cdots {\epsilon }_{{\alpha }_s}^{-1},$$

$${\epsilon }_{\alpha }=|{\lambda }^{\alpha }-{\lambda }_{i_{\alpha }}|.$$

$$n-1\ge 2\underset{1\le i\le d}{\min }|{\lambda }_i|$$

Full text

The author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006

1. 1 Local basins of attraction

2. 2 Elimination of terms

3. 2.1 The one-resonant case

4. 3 Classification of stable orbits

5. 3.1 Global basins are Fatou components

6. 4 Internal dynamics and geometry

7. 4.1 Fatou coordinates

8. 4.2 Orbit behaviour

9. 4.3 Geometry of the global basins

10. 5 Periodic cycles

Introduction

When studying the behaviour of iterates of a holomorphic endomorphism $F$ of $\mathbb{C}^{d}$, $d\geq 1$, one of the basic objects of interest is the Fatou set

[TABLE]

A connected component of $\mathcal{F}$ is called a Fatou component of $F$. Let $V$ be a Fatou component of $F$. Then $V$ is invariant, if $F(V)=V$. More generally, $V$ is $p$-periodic for $p\in\mathbb{N}^{*}$, if $F^{p}(V)=V$ and $F^{q}(V)\neq V$ for $q<p$. In this case we call $(V,F(V),\cdots,F^{p-1}(V))$ a $p$-periodic cycle of Fatou components. A Fatou component $V$ is attracting to $P\in\mathbb{C}^{d}$, if $(F|_{V})^{n}\to P$ (then in particular $F(P)=P$). A periodic Fatou component $V$ attracting to $P$ is recurrent if $P\in V$ and non-recurrent if $P\in\partial V$.

Every recurrent attracting Fatou component of an automorphism of $\mathbb{C}^{d}$ is biholomorphic to $\mathbb{C}^{d}$ (this follows from [PVW08, Theorem 2] and the appendix of [RR88]). For polynomial automorphisms of $\mathbb{C}^{2}$, even non-recurrent attracting periodic Fatou components are biholomorphic to $\mathbb{C}^{2}$ (by [LP14, Theorem 6] and [Ued86]).

In [BRS], F. Bracci, J. Raissy and B. Stensønes proved the existence of automorphisms of $\mathbb{C}^{d}$ with a non-recurrent attracting invariant Fatou component biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. In particular this provided first examples of automorphisms of $\mathbb{C}^{2}$ with a multiply connected attracting Fatou component (those are necessarily non-polynomial by the previously mentioned results). Based on this, it is easy to construct automorphisms of $\mathbb{C}^{d}$ with non-recurrent attracting invariant Fatou components biholomorphic to $\mathbb{C}^{d-m}\times(\mathbb{C}^{*})^{m}$ for $m<d$ (see Corollary 5).

By [Ued86, Proposition 5.1], attracting Fatou components are Runge, and, by [Ser55], for every Runge domain $D\subseteq\mathbb{C}^{d}$, we have $H^{q}(D)=0$ for $q\geq d$. Hence $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ has the highest possible degree of non-vanishing cohomology for an attracting Fatou component. It is an open question whether all non-recurrent attracting invariant Fatou components of automorphisms are biholomorphic to a product of copies of $\mathbb{C}$ and $\mathbb{C}^{*}$. To the author’s knowledge it is not even clear these are the only homotopy types that can occur.

Non-recurrent attracting Fatou components of type $\mathbb{C}^{d}$ appear in parabolic flowers (generalisations of one-dimensional Leau-Fatou flowers), that is in arbitrary finite number around a fixed point and grouped in periodic cycles. In this paper we generalise the example of [BRS] to higher orders to show that the same can occur for type $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. We further extend their results to provide a complete classification of stable orbits near the fixed point in these examples.

We will be studying germs $F$ of automorphisms of $\mathbb{C}^{d}$ at the origin of the form

[TABLE]

where $\lambda_{1},\ldots,\lambda_{d}$ are of unit modulus, not roots of unity, such that $F$ is one-resonant via $\lambda_{1}\cdots\lambda_{d}=1$, i.e. $\lambda_{1}^{m_{1}}\cdots\lambda_{d}^{m_{d}}=\lambda_{j}$ for $m_{1},\ldots,m_{d}\in\mathbb{N}$ and $j\in\{1,\ldots,d\}$ if and only if $(m_{1},\ldots,m_{d})=(q,\ldots,q)+e_{j}$ for some $q\in\mathbb{N}$ (see Definition 1.1), and $l>2kd+1$. In some parts we will in addition assume all subsets $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$, $j=1,\ldots,d$ to satisfy the Brjuno condition (Definition 2.4). For $k=1$ this is precisely the set-up of [BRS].

Our main results are the following:

Theorem 1**.**

Let $F$ be a germ of automorphisms of $\mathbb{C}^{d}$ at the origin of the form (1). Then $F$ admits $k$ disjoint, completely invariant ($F(\Omega_{h})=\Omega_{h}=F^{-1}(\Omega_{h})$), attracting basins $\Omega_{0},\ldots,\Omega_{k-1}$ such that

1. (1)

If each subset $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$, $j=1,\ldots,d$ satisfies the Brjuno condition, then:

1. (a)

$\Omega_{h}$* is a union of Fatou components for each $h=0,\ldots,k-1$,* 2. (b)

$F$* admits Siegel hypersurfaces (i.e. invariant hypersurfaces on which $F$ acts as a rotation) tangent to each coordinate hyperplane,* 3. (c)

All stable orbits of $F$ near the origin are contained in one of the above. 2. (2)

If $F$ is a global automorphisms of $\mathbb{C}^{d}$, then for each $h=0,\ldots,k-1$ there exists a biholomorphic map $\phi_{h}:\Omega_{h}\to\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ conjugating $F$ to

[TABLE]

Moreover, there exist automorphisms of the form (1) for each admissible choice of $\lambda_{1},\ldots,\lambda_{d}$ and $l>2kd+1$.

Remark 2*.*

Each global basin $\Omega_{h}$ arises as the union of all iterated preimages of an explicit local attracting basin $B_{h}$ of the desired homotopy type whose external geometry becomes apparent in polar decomposition as depicted in Figure 1 for $d=k=2$. The global basins are more abstract, so we don’t know much about their outer shape or arrangement.

Remark 3*.*

Each attracting orbit in a basin $\Omega_{h}$ converges tangent to a real $d$-dimensional submanifold (depending on the orbit), but not tangent to any complex subspace.

Theorem 4**.**

Let $p\in\mathbb{N}^{*}$ divide $k$. Then there exist automorphisms $G$ of $\mathbb{C}^{d}$ such that $G^{p}$ has the form (1) and $G(\Omega_{h})=\Omega_{h+p\bmod k}$ for $h=0,\ldots,k-1$. In particular, $G$ admits $k/p$ disjoint $p$-cycles of non-recurrent, attracting Fatou components biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$, that are all attracted to the origin.

As an immediate corollary, we obtain automorphisms with cycles of non-recurrent attracting Fatou components biholomorphic to any product of copies of $\mathbb{C}$ and $\mathbb{C}^{*}$ with admissible cohomology:

Corollary 5**.**

Let $d,k\in\mathbb{N}^{*}$, $p\in\mathbb{N}^{*}$ divide $k$, and $0\leq m<d$. Then there exist holomorphic automorphisms of $\mathbb{C}^{d}$ possessing $k/p$ disjoint $p$-cycles of non-recurrent, attracting, invariant Fatou components biholomorphic to $\mathbb{C}^{d-m}\times(\mathbb{C}^{*})^{m}$ and attracted to the origin.

We also prove an auxiliary result on holomorphic elimination of infinite families of monomials that may be interesting in its own right. We use multi index notation $\lambda^{\alpha}=\lambda_{1}^{\alpha_{1}}\cdots\lambda_{d}^{\alpha_{d}}$ for $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{N}^{d}$ and define the notion of a Brjuno set of exponents $A\subseteq\mathbb{N}^{d}$, by requiring a Brjuno condition only on the small divisors $\lambda^{\alpha}-\lambda_{j}$ with $\alpha\in A$ and $j\in\{1,\ldots,d\}$ (see Definition 2.2).

Theorem 6**.**

Let $F$ be a germ of endomorphisms of $\mathbb{C}^{d}$ of the form $F(z)=\Lambda z+\sum_{|\alpha|>1}\sum_{j=1}^{d}f_{\alpha}^{j}z^{\alpha}e_{j}$ with $\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$. Let $A_{0}$ and $A$ be disjoint sets of multi-indices in $\mathbb{N}^{d}$ such that $A$ admits a partition $A=A_{1}\cup\cdots\cup A_{k_{0}}$ such that

1. (1)

For $0\leq k\leq k_{0}$, if $\alpha\in A_{k}$ and $\beta\leq\alpha$, then $\beta\in A_{\overline{k}}:=A_{0}\cup\cdots\cup A_{k}$ (where $\leq$ is taken component-wise). 2. (2)

For $1\leq k\leq k_{0}$, if $\beta_{1},\ldots,\beta_{l}\in A_{0}$ such that $\beta_{1}+\cdots+\beta_{l}\in A_{\overline{k}}$, $|\beta_{1}|\geq 2$, and $f_{\beta_{1}}^{j_{1}}\cdots f_{\beta_{l}}^{j_{l}}\neq 0$, then $e_{j_{1}}+\cdots+e_{j_{l}}\notin A_{k}$. 3. (3)

$A$* is a Brjuno set for $F$.*

Then there exists a local biholomorphism $H\in\operatorname{Aut}(\mathbb{C}^{d},0)$ conjugating $F$ to $G=H^{-1}\circ F\circ H$ where $G(z)=\sum_{|\alpha|>1}g_{\alpha}z^{\alpha}$ with $g_{\alpha}=f_{\alpha}$ for $\alpha\in A_{0}$ and $g_{\alpha}=0$ for $\alpha\in A$.

The proof of the theorem is based on that of a partial linearisation result from [Pös86] which it generalises.

Outline

In Section 1, following [BRS], we recall results from [BZ13] that show that germs of the form (1) have $k$ local attracting basins of the desired homotopy type. We then examine their arrangement in the surrounding space.

In Section 2 we prove Theorem 6 and, under the aforementioned Brjuno-type condition, we conclude the existence of local coordinates that allow us to better control the unknown tail of $F$.

In Section 3 we use those coordinates to extend [BRS, Lemma 2.5] to classify the stable orbits of $F$ near the origin, proving the first part of Theorem 1.

In Section 4 we define two closely related systems of coordinates on each local basin compatible with the action of $F$: the first, in a small variation of [BRS, Section 3], allows us to study the behaviour of attracting orbits more carefully in Section 4.2, showing Remark 3; the second in Section 4.3 conjugates $F$ to an affine map and, if $F$ is an automorphism, extends to a biholomorphism from the corresponding global basin to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. The existence of automorphisms of the form (1) follows directly from a jet-interpolation result, concluding the proof of the second part of Theorem 1.

Finally, we show Theorem 4 and Corollary 5 in Section 5 via explicit calculations.

Conventions

$\operatorname{Aut}(\mathbb{C}^{d})$ is the set of biholomorphic automorphisms of $\mathbb{C}^{d}$ and $\operatorname{Aut}(\mathbb{C}^{d},0)$ the set of germs of biholomorphisms of $\mathbb{C}^{d}$ at the origin such that $F(0)=0$.

For $z\in\mathbb{C}^{d}$ and $F\in\operatorname{Aut}(\mathbb{C}^{d},0)$, upper indices denote the components of $z=(z^{1},\ldots,z^{d})$, while a lower index $n\in\mathbb{N}:=\{0,1,\ldots\}$ denotes the iterated image $z_{n}=(z_{n}^{1},\ldots,z_{n}^{d}):=F^{n}(z)$ of $z$ under $F$. Similarly, for the coordinates $u:=\pi(z):=z^{1}\cdots z^{d}$, $U:=u^{-k}$ we set $u_{n}:=\pi(z_{n})$ and $U_{n}:=u_{n}^{-k}$.

For $\{x_{n}\}_{n\in\mathbb{N}}$ a sequence of objects, we say the object $x_{n}$ has a property eventually, if there exists $n_{0}\in\mathbb{N}$ such that $x_{n}$ has this property for all $n>n_{0}$.

For a topological space $D$ and maps $f,g:D\to\mathbb{C}$ , we use Bachmann-Landau notation for global behaviour:

• •

$f(x)=O(g(x))$ for $x\in D$, if $|f(x)|\leq C|g(x)|$ for all $x\in D$ for some $C>0$,

• •

$f(x)\approx g(x)$ for $x\in D$, if $f(x)=O(g(x))$ and $g(x)=O(f(x))$ (often denoted $f(x)=\Theta(g(x))$),

and for asymptotic behaviour:

• •

$f(x)=O(g(x))$ as $x\to x_{0}$, if $\limsup_{x\to x_{0}}\frac{|f(x)|}{|g(x)|}=C<+\infty$,

• •

$f(x)\approx g(x)$ as $x\to x_{0}$, if $f(x)=O(g(x))$ and $g(x)=O(f(x))$ as $x\to x_{0}$,

• •

$f(x)=o(g(x))$ as $x\to x_{0}$, if $\lim_{x\to x_{0}}\frac{|f(x)|}{|g(x)|}=0$,

• •

$f(x)\sim g(x)$ as $x\to x_{0}$, if $\lim_{x\to x_{0}}\frac{f(x)}{g(x)}=1$ or $f(x)=g(x)(1+o(1))$ as $x\to x_{0}$.

1. Local basins of attraction

After recalling a construction of local basins of attraction, we give their representation in internal holomorphic coordinates to determine their homotopy type, and in external polar coordinates to visualise their arrangement in $\mathbb{C}^{d}$.

The local basins arise from the study of local dynamics of one-resonant germs in [BZ13] by F. Bracci and D. Zaitsev.

Definition 1.1**.**

A germ $F$ of endomorphisms of $\mathbb{C}^{d}$ at the origin such that $F(0)=0$ and $dF_{0}=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$ is called one-resonant of index $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{N}^{d}$, if $\lambda_{j}=\lambda_{1}^{m_{1}}\cdots\lambda_{d}^{m_{d}}$ for some $j\leq d$ and $m=(m_{1},\ldots,m_{d})\in\mathbb{N}^{d}$ if and only if $m=k\alpha+e_{j}$ for some $k\in\mathbb{N}$ (where $e_{j}$ denotes the $j$-th unit vector).

Remark 1.2*.*

For $1\leq j\leq d$, one-resonance of index $\alpha\neq n\cdot e_{j}$ for every $n\in\mathbb{N}$ implies in particular that $\lambda_{j}$ is not a root of unity.

We start with germs of biholomorphisms of $\mathbb{C}^{d}$ at the origin in normal form $F_{{\rm N}}$ given for $z=(z^{1},\ldots,z^{d})$ by

[TABLE]

that are one-resonant of index $(1,\ldots,1)$ with $\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$ such that $|\lambda_{j}|=1$ for each $j\leq d$. We will later moreover assume that proper subsets of $\{\lambda_{1},\ldots,\lambda_{d}\}$ satisfy the Brjuno condition (see Definition 2.4).

An important tool to study the dynamics of this type of maps introduced in [BZ13] is the variable $u=\pi(z)=z^{1}\cdots z^{d}$ on which $F_{{\rm N}}$ acts parabolically of order $k$ near the origin as $u\mapsto u(1-u^{k})+O(u^{2k+1})$, yielding a Leau-Fatou flower of $k$ attracting sectors

[TABLE]

for $h=0,\ldots,k-1$ and suitable $R>0$ and $\theta\in(0,\nicefrac{{\pi}}{{2k}})$. Note that on each such sector the map $u\mapsto u^{-k}=:U$ is injective, hence each sector is biholomorphic to a “sector at infinity”

[TABLE]

To control $z$ in terms of $u=\pi(z)$, for $\beta\in(0,1/d)$ let further

[TABLE]

and for $h=0,\ldots,k-1$

[TABLE]

Now from the proof of [BZ13, Theorem 1.1] it follows:

Theorem 1.3**.**

Let $F_{{\rm N}}$ be of the form (1.1) and $l\in\mathbb{N}$, $l>2kd+1$. Then for every germ $F$ of automorphisms of $\mathbb{C}^{d}$ at the origin of the form

[TABLE]

for every $\beta_{0}\in(0,1/d)$ such that $\beta_{0}(l+d-1)>2k+1$, and every $\theta_{0}\in(0,\nicefrac{{\pi}}{{2k}})$, there exists $R_{0}>0$ such that the (disjoint, non-empty) open sets $B_{h}:=B_{h}(R_{0},\theta_{0},\beta_{0})$ for $h=0,\ldots,k-1$ are uniform local basins of attraction for $F$, that is $F(B_{h})\subseteq B_{h}$, and $\lim_{n\to\infty}F^{n}\equiv 0$ uniformly in $B_{h}$ for each $h$.

Remark 1.4*.*

As in [BRS, Lemma 2.7 and Section 7], we observe that each local basin $B_{h}$ is homotopy equivalent to $(S^{1})^{d-1}$, so the local basins have the desired homotopy type (of $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$).

To see this, let again $u=z^{1}\cdots z^{d}$. Then $(u,z^{\prime})=(u,z^{2},\ldots,z^{d})$ is a holomorphic system of coordinates on $B_{h}$ through which $B_{h}$ is biholomorphic to

[TABLE]

Since the sector $S_{h}(R_{0},\theta_{0})$ is contractible, and for each $j$ given $u,z^{2},\ldots,z^{j}$, the value of $z^{j+1}$ is confined to an annulus, this is homotopy equivalent to $(S^{1})^{d-1}$.

Remark 1.5* (Shape and arrangement).*

The external shape of the local basins becomes apparent in polar coordinates. For the sake of visualisation and simplicity, let $d=2$ and assume $0<R<1$. To get a global (real) smooth argument coordinate, we consider $\arg$ to take values in $S^{1}=\mathbb{R}/(2\pi\mathbb{Z})$, so $(\arg(z),\arg(w))$ is a point on the torus $\mathbb{T}^{2}=S^{1}\times S^{1}$. For $0<R<1$ the condition {\big{\lvert}u^{k}-\frac{1}{2R}\big{\rvert}}<\frac{1}{2R} is implied by the others and so for $h\in\{0,\ldots,k-1\}$, we have

[TABLE]

In this case $B_{h}$ is diffeomorphic via polar coordinates $(|z|,|w|,\arg(z),\arg(w))$ to the product

[TABLE]

shown in Figure 1. The modulus component is simply connected and identical for all basins and the argument component of $B_{h}$ is a $\theta$-neighbourhood of the central curve $s+t\equiv 2\pi h/k$, that is a “ribbon” winding around the torus $\mathbb{T}^{2}$.

For $d>2$, the modulus component $W(\beta)\cap\mathbb{R}_{+}^{d}$ is still simply connected and the argument component of $B_{h}$ is a $\theta$-neighbourhood of a central hypersurface in $\mathbb{T}^{d}$ given by $s_{1}+\cdots+s_{d}\equiv 2\pi h/k$. For general $R>0$, the basins are truncated, but remain the same near the origin and preserve their homotopy type.

2. Elimination of terms

In this section, we will prove that under a Brjuno-type condition we can holomorphically eliminate infinite families of monomials even in the presence of resonances. We will then apply this to germs of the form (1.2) to simplify the unknown tail.

Throughout this section, we will use multi-index notation:

Notation 2.1*.*

Let $\alpha=(\alpha_{1},\ldots,\alpha_{d})\in\mathbb{N}^{d}$ be a multi-index and $z=(z^{1},\ldots,z^{d})\in\mathbb{C}^{d}$. Then we set $z^{\alpha}:=(z^{1})^{\alpha_{1}}\cdots(z^{d})^{\alpha_{d}}$ and $|\alpha|:=\alpha_{1}+\cdots+\alpha_{d}$. For $\alpha,\beta\in\mathbb{N}^{d}$, we write $\alpha\leq\beta$ if $\alpha_{j}\leq\beta_{j}$ for $j=1,\ldots,d$.

We first introduce the notion of a Brjuno set of exponents:

Definition 2.2**.**

Let $F$ be a germ of endomorphisms of $\mathbb{C}^{d}$ with

[TABLE]

A set $A\subseteq\mathbb{N}^{d}$ is a Brjuno set (of exponents) for $(\lambda_{1},\ldots,\lambda_{d})$ (or for $F$), if

[TABLE]

where

[TABLE]

for $k\geq 2$.

Remark 2.3*.*

Subsets and finite unions of Brjuno sets are Brjuno sets.

This definition includes the classical Brjuno condition from [Brj73] and the partial Brjuno condition from [Pös86] as follows:

Definition 2.4**.**

Let $\lambda_{1},\ldots,\lambda_{d}\in\mathbb{C}$.

1. (1)

$\{\lambda_{1},\ldots,\lambda_{d}\}$ satisfies the Brjuno condition, if $A=\mathbb{N}^{d}$ is a Brjuno set for $(\lambda_{1},\ldots,\lambda_{d})$. 2. (2)

$L\subseteq\{\lambda_{1},\ldots,\lambda_{d}\}$ satisfies the partial Brjuno condition (wrt. $(\lambda_{1},\ldots,\lambda_{d})$), if $A=\{\alpha\in\mathbb{N}^{d}\mid\alpha_{j}=0\text{ for }\lambda_{j}\notin L\}$ is a Brjuno set for $(\lambda_{1},\ldots,\lambda_{d})$.

[Brj73] and [Pös86] prove full and partial analytic linearisability on submanifolds tangent to the union of the eigenspaces of the multipliers that satisfy the respective condition. The following theorem generalises these results in the context of eliminating infinite families of monomials. A different generalisation aiming at full linearisation in the presence of resonances has been explored in [Rai11].

Theorem 2.5**.**

Let $F$ be a germ of endomorphisms of $\mathbb{C}^{d}$ of the form $F(z)=\Lambda z+\sum_{|\alpha|>1}\sum_{j=1}^{d}f_{\alpha}^{j}z^{\alpha}e_{j}$ with $\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$. Let $A_{0}$ and $A$ be disjoint sets of multi-indices in $\mathbb{N}^{d}$ such that

1. (1)

If $\alpha\in A_{0}$ and $\beta\leq\alpha$, then $\beta\in A_{0}$, and if $\alpha\in A$ and $\beta\leq\alpha$, then $\beta\in A_{0}\cup A$. 2. (2)

If $\beta_{1},\ldots,\beta_{l}\in A_{0}$ and $\beta_{1}+\cdots+\beta_{l}\in A_{0}\cup A$, $|\beta_{1}|\geq 2$ and $f_{\beta_{1}}^{j_{1}}\cdots f_{\beta_{l}}^{j_{l}}\neq 0$, then $e_{j_{1}}+\cdots+e_{j_{l}}\notin A$. 3. (3)

$A$* is a Brjuno set for $F$.*

Then there exists a local biholomorphism $H\in\operatorname{Aut}(\mathbb{C}^{d},0)$ conjugating $F$ to $G=H^{-1}\circ F\circ H$ such that $G(z)=\sum g_{\alpha}z^{\alpha}$ with $g_{\alpha}=f_{\alpha}$ for $\alpha\in A_{0}$ and $g_{\alpha}=0$ for $\alpha\in A$.

For $A_{0}=\{|\alpha|\leq 1\}$ and $A=\mathbb{N}^{d}\backslash A_{0}$ or $A=(\mathbb{N}^{m}\times\{0\})\backslash A_{0}$, we recover the results from [Brj73] and [Pös86]. A novelty of phrasing the result in this way is that it can be iterated to obtain Theorem 6.

Remark 2.6*.*

If we assume $\sum_{\alpha\in A_{0}}f_{\alpha}z^{\alpha}$ to be in Poincaré-Dulac normal form, the condition $f_{\beta_{1}}^{j_{1}}\cdots f_{\beta_{l}}^{j_{l}}\neq 0$ can be replaced by $\lambda^{\beta_{m}}\neq\lambda_{j_{m}}$ for $1\leq m\leq l$ to avoid dependence of Condition (2) on the specific germ $F$.

The proof of Theorem 2.5 emerges largely by careful examination of that in [Pös86] with some adjustments to avoid the assumption $\min_{1\leq j\leq d}|\lambda_{j}|\leq 1$ in the proofs of Lemmas 2.7 and 2.8. In [Pös86] this is ensured by considering $F^{-1}$ if necessary, but Condition (2) in our theorem is not invariant under taking inverses.

Proof.

Formal series

[TABLE]

of the required form emerge as solutions to the homological equation $F\circ H=H\circ G$. Comparing coefficients for $\alpha\in\mathbb{N}^{d}\backslash\{0\}$, this means

[TABLE]

where $e_{J}:=e_{j_{1}}+\cdots+e_{j_{k}}$. Take $h_{\alpha}=0$ for $\alpha\notin A$. Then for $\alpha\in A_{0}$, the first term in the sum vanishes by Condition (1) and the second term vanishes by Condition (2), so $g_{\alpha}=f_{\alpha}$. For $\alpha\in A$, $\lambda^{\alpha}\operatorname{id}-\Lambda$ is invertible by Condition (3) and the right hand side depends only on $h$-terms with index of order less than $|\alpha|$. Hence (2.3) determines $h_{\alpha}$ uniquely by recursion and we obtain a formal solution $H$ and hence $G=H^{-1}\circ F\circ H$.

To show that $H$ (and hence $G$) converges in some neighbourhood of the origin, we have to show

[TABLE]

We apply the majorant method first used by C. L. Siegel in [Sie42] and improved in [Brj73]. We may assume (up to scaling of variables) that $\lVert f_{\alpha}\rVert_{1}\leq 1$ for all $|\alpha|\geq 2$. Now for $\alpha\in A$ again by Condition (2), the second term in the sum in (2.3) vanishes and it follows

[TABLE]

where $\varepsilon_{\alpha}:=\min_{1\leq i\leq d}|\lambda^{\alpha}-\lambda_{i}|$. We estimate (2.5) in two parts, one on the number of summands, the other on their size. We define recursively $\sigma_{1}=1$ and

[TABLE]

and $\delta_{e_{1}}=\cdots=\delta_{e_{d}}=1$, $\delta_{\alpha}=0$ for $\alpha\notin A\cup\{e_{1},\ldots,e_{d}\}$, and

[TABLE]

Then, by induction on $|\alpha|$, (2.5) implies

[TABLE]

for $\alpha\in A$. Hence to prove (2.4) it suffices to prove estimates of the same type for $\sigma_{|\alpha|}$ and $\delta_{\alpha}$.

The estimates on $\sigma_{r}$ go back to [Sie42] and [Ste61]: Let $\sigma(t):=\sum_{r=1}^{\infty}\sigma_{r}t^{r}$ and observe

[TABLE]

Solving for $t$ and requiring $\sigma(0)=0$ yields a unique holomorphic solution

[TABLE]

for small $t$, so $\sigma$ converges near [math] and we have

[TABLE]

The estimates on $\delta_{\alpha}$ take care of the small divisors $\varepsilon_{\alpha}$ and proceed essentially like [Brj73]. For every $|\alpha|\geq 2$, we choose a maximising decomposition $\beta_{1}+\cdots+\beta_{k}=\alpha$ in (2.7) such that

[TABLE]

and $|\alpha|>|\beta_{1}|\geq\cdots\geq|\beta_{k}|\geq 1$. In this way, starting from $\delta_{\alpha}$ we proceed to decompose $\delta_{\beta_{1}},\ldots,\delta_{\beta_{k}}$ in the same way and continue the process until we arrive at a well-defined decomposition of the form

[TABLE]

where $2\leq|\alpha_{s}|,\ldots,|\alpha_{1}|<|\alpha_{0}|$ and $\alpha_{0}=\alpha$. We may further choose an index $i_{\alpha}\in\{1,\ldots,d\}$ for each $|\alpha|\geq 2$ such that

[TABLE]

Let $n\in\mathbb{N}$ such that

[TABLE]

and $\theta>0$ such that

[TABLE]

Now for the indices $\alpha_{0},\ldots,\alpha_{s}$ in the decomposition (2.11), we want to bound

[TABLE]

for $j\leq d$ and $m\in\mathbb{N}$, where we adopt the convention $\omega_{A}(1):=+\infty$. First we need the following lemma, that [Pös86] attributes to Siegel, showing that indices contributing to $N_{m}^{j}(\alpha)$ cannot be to close to each other:

Lemma 2.7** (Siegel).**

Let $m\geq 1$. If $\alpha>\beta$ are such that $\varepsilon_{\alpha}<\theta\omega_{A}(m),$ $\varepsilon_{\beta}<\theta\omega_{A}(m),$ and $i_{\alpha}=i_{\beta}=j,$ then $|\alpha-\beta|\geq m$.

Proof.

For $m=1$, $\alpha>\beta$ implies $|\alpha-\beta|\geq 1$. For $m\geq 2$, by the definition of $\omega_{A}$ in (2.2), we have $\omega_{A}(m)\leq 1$. With that and (2.12), the hypothesis $\varepsilon_{\beta}<\theta\omega_{A}(m)$ implies

[TABLE]

and hence

[TABLE]

i.e. $\omega(m)>\omega(|\alpha-\beta|+1)$. But $\omega$ is decreasing, so we must have $|\alpha-\beta|\geq m$.∎

We can now show Brjuno’s estimate on $N_{m}^{j}(\alpha)$:

Lemma 2.8** (Brjuno, [Brj73]).**

For $|\alpha|\geq 2$, $m\geq 1$, and $1\leq j\leq d$, we have

[TABLE]

Proof.

We fix $m$ and $j$ and proceed by induction on $|\alpha|$.

If $2\leq|\alpha|\leq m$, we have

[TABLE]

for all $0\leq l\leq s$, so $N_{m}^{j}(\alpha)=0.$

If $|\alpha|>m$, we take the chosen decomposition (2.10) and note that only $|\beta_{1}|$ may be greater than $K=\max\{|\alpha|-m,m\}$. If $|\beta_{1}|>K$, we decompose $\delta_{\beta_{1}}$ in the same way and repeat this at most $m-1$ times to obtain a decomposition

[TABLE]

with $0\leq k\leq m-1$, $l\geq 2$ and

[TABLE]

In particular, (2.14) implies $|\alpha-\alpha_{k}|<m$. Hence Lemma 2.7 shows that at most one of the $\varepsilon$- factors in (2.13) can contribute to $N_{m}^{j}(\alpha)$ and we have

[TABLE]

Now let $0\leq h\leq l$ such that be the such that $|\beta_{1}|,\ldots,|\beta_{h}|>m\geq|\beta_{h+1}|,\ldots,|\beta_{l}|$. Then by (2.14), we have $|\beta_{1}|,\ldots,|\beta_{h}|\leq|\alpha|-m$ and, by induction, the terms with $|\beta|\leq m$ vanish and we have

[TABLE]

To estimate the product (2.11) we partition the indices into sets

[TABLE]

(recall for $I_{0}$ the convention $\omega_{A}(1)=+\infty$). By Lemma 2.8, we have

[TABLE]

and we can estimate

[TABLE]

This bound is independent of $\alpha$ and, since $A$ is a Brjuno set, it is finite. Hence with (2.8) and (2.9) it follows that

[TABLE]

and thus $H$ and $G$ converge. ∎

Proof of Theorem 6.

Assume by induction on $k_{0}$, that $f_{\alpha}=0$ for $\alpha\in A\backslash A_{k_{0}}$. We show that $A_{0}^{\prime}=A_{\overline{k_{0}-1}}$ and $A^{\prime}=A_{k_{0}}$ satisfy the prerequisites of Theorem 2.5.

Conditions (1) and (3) follow directly from their counterparts.

Let $\beta_{1},\ldots,\beta_{l}\in A_{0}^{\prime}=A_{\overline{k_{0}-1}}$ as in Condition (2). By induction $f_{\beta_{1}}^{j_{1}}\cdots f_{\beta_{l}}^{j_{l}}\neq 0$ implies $\beta_{1},\ldots,\beta_{l}\in A_{0}$, so Assumption (2) of Theorem 6 implies $e_{J}\notin A_{k}=A^{\prime}$, and Condition (2) is satisfied.

Therefore Theorem 2.5 shows that $F$ is conjugate to $G$ with $g_{\alpha}=f_{\alpha}$ for $\alpha\in A_{0}$ and $g_{\alpha}=0$ for $\alpha\in A$. ∎

2.1. The one-resonant case

In the one-resonant case, the classical Brjuno condition on subsets already implies much more:

Lemma 2.9**.**

If $F\in\operatorname{Aut}(\mathbb{C}^{d},0)$ is one-resonant of index $\alpha=(1,\ldots,1)$ at [math], then the following are equivalent:

1. (1)

$\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$* satisfies the Brjuno condition for every $j\leq d$.* 2. (2)

$\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$* satisfies the partial Brjuno condition wrt. $(\lambda_{1},\ldots,\lambda_{d})$ for every $j\leq d$.* 3. (3)

$A_{k}=\{\beta\in\mathbb{N}^{d}\mid|\beta|>kd+1,\min\{\beta_{1},\ldots,\beta_{d}\}=k\}$* is a Brjuno set for $F$ for every $k\in\mathbb{N}$.*

Proof.

The relevant minimal divisors for Items (1) and (2) are

[TABLE]

and $\omega_{M_{j}}(l)$ with $M_{j}:=\{\beta\in\mathbb{N}^{d}\mid\beta_{j}=0\}$, respectively, for $j\leq d$ and $l\geq 2$.

**(3) $\Rightarrow$ (2) $\Rightarrow$ (1) follows from $\omega_{A_{0}}\leq\omega_{M_{j}}\leq\omega_{j}$.

(1) $\Rightarrow$ (2). Fix $j\leq d$ and let $m_{0}\geq 2$ large enough that $2^{m_{0}}\geq d-1$. Then for $m\geq m_{0}$ the only divisors contributing to $\omega_{M_{j}}(2^{m})$, but not to $\omega_{j}(2^{m})$, are of the form

[TABLE]

where $\theta:=|\lambda_{j}|\max_{i\neq j}|\lambda_{i}|^{-1}$. Hence, we have

[TABLE]

If we have (1) this is finite for each $j\leq d$, implying (2).

**(2) $\Rightarrow$ (3). For $\beta\in A_{k}$, we have $|\beta-k\alpha|\geq 2$ and $\beta_{j}=k$ for some $j\leq d$, so $(\beta-k\alpha)_{j}=0$ and for any $i\leq d$, we have

[TABLE]

Hence

[TABLE]

If we have (2), this is finite, implying (3). ∎

With this, we only need the weakest assumption (1) to show that we can assume the tail of our map to be of a nicer form:

Corollary 2.10**.**

Let $F$, $\lambda_{1},\ldots,\lambda_{d}$ and $l\in\mathbb{N}$ be as in Theorem 1.3 such that $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$ satisfies the Brjuno condition for every $j\leq d$, and let $\alpha=(1,\ldots,1)$. Then there exist a local change of coordinates $\chi(z)=z+O(\lVert z\rVert^{l})$ conjugating $F$ to

[TABLE]

Proof.

First, observe that, since $l>2k+1$, [BZ13, Theorem 3.6] implies that $F$ is conjugate to $G$ with $G(z)=F_{{\rm N}}(z)+O(\lVert z\rVert^{ld+2})$. We want to apply Theorem 6 to $G$, $A_{0}=\{|\beta|\leq ld+1\}$ and

[TABLE]

for $1\leq m\leq l+1$. Condition (1) is clear. For $\beta\in A_{0}$ with $f_{\beta}^{j}\neq 0$, we have $\beta=\epsilon\alpha+e_{j}$ with $\epsilon=0,1$, so $\beta\geq e_{j}$ and $\beta\geq\alpha+e_{j}$ if $|\beta|\geq 2$. Hence for $\beta_{1},\ldots,\beta_{r}\in A_{0}$ with $|\beta_{1}|\geq 2$ such that $f_{\beta_{1}}^{j_{1}}\cdots f_{\beta_{r}}^{j_{r}}\neq 0$, we have

[TABLE]

If $\beta_{1}+\cdots+\beta_{r}\in A_{0}$, then this implies $e_{J}\in A_{0}$. If $\beta_{1}+\cdots+\beta_{r}\in A_{m}$, then for some $i\leq d$, we have

[TABLE]

by (2.15). So in both cases, $e_{J}\notin A_{m^{\prime}}$ for any $m^{\prime}\geq\max\{1,m\}$ and Condition (2) is satisfied. Condition (3) follows from Lemma 2.9. Now Theorem 6 shows that $G$ is locally conjugate to $H$ such that $H(z)=F_{{\rm N}}(z)+R(z)$, where $R(z)$ only contains monomials $z^{\beta}$ with $\beta\in\mathbb{N}_{0}^{d}\backslash A_{\overline{l+1}}=\{\beta\geq l\alpha\}$ or $R(z)=O(z^{l\alpha})$. ∎

3. Classification of stable orbits

In this section, under the Brjuno condition on subsets, we identify all stable orbits of our germs near the fixed point and conclude that the global basins corresponding to our local basins are (unions of) Fatou components.

Corollary 2.10 implies immediately that there exist rotating stable orbits that do not converge to the origin:

Corollary 3.1**.**

Let $F\in\operatorname{Aut}(\mathbb{C}^{d},0)$ be as in Theorem 1.3, i.e. $F(z)=F_{{\rm N}}(z)+O(\lVert z\rVert^{l})$, where

[TABLE]

Assume further that $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$ satisfies the Brjuno condition for every $j\leq d$. Then $F$ admits Siegel hypersurfaces $D_{1},\ldots,D_{d}$ tangent to $\{z_{1}=0\},\cdots,\{z_{d}=0\}$ respectively.

Proof.

Let $(\chi_{1},\ldots,\chi_{d})(z)=\chi(z)=z+O(\lVert z\rVert^{l})$ be as in Corollary 2.10. Then on $D_{j}:=\{\chi_{j}(z)=0\}$ for $j=1,\ldots,d$, $F$ acts as the irrational rotation $w\mapsto\Lambda w$. ∎

In fact, using Corollary 2.10, we can extend the proof of [BRS, Lemma 2.5] to classify all stable orbits near the origin:

Proposition 3.2**.**

Let $F$ and $B_{1},\ldots,B_{k-1}$ be as in Theorem 1.3 such that $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$ satisfies the Brjuno condition for every $j\leq d$. For $z\in\mathbb{C}^{d}$ let $z_{n}:=F^{\circ n}(z)$ and $u_{n}:=z_{n}^{\alpha}$. Then there exists $r>0$ such that: if $z_{n}\in B_{r}(0)$ eventually, then either $\{z_{n}\}_{n}$ is contained in one of the Siegel hypersurfaces $D_{1},\ldots,D_{d}$, or $z_{n}\to 0$ and there exists a unique $h\in\{0,\ldots,k-1\}$ such that

1. (1)

u_{n}\sim e^{2\pi ih/k}n^{-{\scriptstyle 1}/\raisebox{-1.0pt}{{\scriptstyle k}}}* (i.e. $\lim_{n\to\infty}\sqrt[k]{n}u_{n}=e^{2\pi ih/k}$), in particular, $|u_{n}|\sim n^{-1/k}$,* 2. (2)

|z_{n}^{j}|\approx n^{-{\scriptstyle 1}/\raisebox{-1.0pt}{{\scriptstyle kd}}}* for $j=1,\ldots,d$,* 3. (3)

for every $R>0,\theta\in(0,\nicefrac{{\pi}}{{2k}}),$ and $\beta\in(0,1/d)$ with $\beta(l+1)>1$, we have $z_{n}\in B_{h}(R,\theta,\beta)$ eventually (in particular, $z_{n}\in B_{h}$ eventually). 4. (4)

The upper bounds $|u_{n}|\leq n^{-1/k}(1+o(1))$ and |z_{n}^{j}|=O(n^{-{\scriptstyle 1}/\raisebox{-1.0pt}{{\scriptstyle kd}}}) in (1) and (2) are uniform in $B_{h}$.

Remark 3.3*.*

In particular, Part (4) shows that ${\{F^{n}\}}_{n\in\mathbb{N}}$ is normal on each local basin $B_{h}$, $h=0,\ldots,k-1$, hence $B_{h}$ is contained in a Fatou component for $F$.

In the proof, we will use the following result of [BRS, Lemma 5.3]:

Lemma 3.4**.**

Let $\beta\in(0,1/d)$ and $l\in\mathbb{N}$ such that $\beta(l+1)>1$. For every germ of biholomorphisms $\chi\in\operatorname{Aut}(\mathbb{C}^{d},0)$ with $\chi(z)=z+O(\lVert z\rVert^{l})$ and every $\beta^{\prime}\in(0,\beta)$, there exists $\varepsilon>0$ such that

[TABLE]

Proof of Proposition 3.2.

First assume

[TABLE]

Then $u_{n}=0$ for some $n\in\mathbb{N}$, if and only if $z_{n}\in D_{j}=\{z_{j}=0\}$ for some $j$ and hence the whole orbit is contained in $D_{j}$. Now assume $u_{n}\neq 0$ for all $n\in\mathbb{N}$ and we can define $U:=u^{-k}$ and $U_{n}:=u_{n}^{-k}$. Then

[TABLE]

Since $l>2k+1$, there exists $r>0$ such that for $z_{n}\in B_{r}(0)$, we have

[TABLE]

So whenever $z_{n}\in B_{r}(0)$ eventually, we have $|U_{n}|\to\infty$. Hence in this case (3.1) shows that for any $c>1$ there exists $n_{c}\in\mathbb{N}$ such that $|U_{n+1}-U_{n}-1|<(c-1)/c$ for all $n\geq n_{c}$. By induction for all $n\geq n_{c}$, we have

[TABLE]

and

[TABLE]

For $c\searrow 1$, it follows

[TABLE]

hence

[TABLE]

Fix $j\in\{1,\ldots,d\}$. By induction on $n\geq 1$ we have

[TABLE]

where $R^{j}(z)=O(z^{\alpha l})=O(u^{l})$. From (3.4), it follows that as $i\to\infty$, we have

[TABLE]

Therefore as $n\to\infty$, we have

[TABLE]

and

[TABLE]

With this and since $R^{j}(z_{i})=O(u_{i}^{l})=O(i^{-l/k})$, (3.5) implies

[TABLE]

since $l>k+1$ and hence $\frac{1}{kd}-\frac{l}{k}<\frac{1}{kd}-1-\frac{1}{k}<-1$. This proves Part (2). ∎

Proof.

Let now $R>0,\theta\in(0,\nicefrac{{\pi}}{{2k}}),\beta\in(0,1/d)$. Then for $n\to\infty$, Part (2) implies

[TABLE]

so $z_{n}\in W(\beta)$ eventually, and by (3.3) we have $U_{n}\in H(R,\theta)$ for large enough $n$. In particular for $(R,\theta,\beta)=(R_{0},\theta_{0},\beta_{0})$, this means $z_{n}\in B_{0}\cup\cdots\cup B_{k-1}$ eventually, but each $B_{h}$ is $F$-invariant by Theorem 1.3, so $z_{n}$ stays in one unique $B_{h}$. Hence $u_{n}$ stays in the image of the unique branch of the $k$-th root centred around \exp\big{\lparen}\frac{2\pi ih}{k}\big{\rparen}, implying Part (3), and we can extract the $k$-th root from (3.4) to get Part (1).

To show Part (4), we recall that in the proof of [BZ13, Theorem 1.1], that implies Theorem 1.3, $R_{0}$, $\theta_{0}$, and $\beta_{0}$ are chosen such that

[TABLE]

Hence in (3.2), $n_{c}$ can be chosen in a uniform manner and, since $\operatorname{Re}U_{n_{c}}>R_{0}$, we get uniform lower bounds on $|U_{n}|>\operatorname{Re}U_{n}$. This becomes a uniform upper bound on the convergence in (3.4) and the subsequent estimates on $|z_{n}|$.

For general $F$, Corollary 2.10 shows that $F$ is locally conjugate to $z\mapsto F_{{\rm N}}(z)+O(z^{l\alpha})$ via a change of coordinates of the form $\chi(z)=z+O(\lVert z\rVert^{l})$. This clearly preserves Part (1) and (2), and by Lemma 3.4 $\chi$ preserves Part (3) and (4) as well. ∎

Remark 3.5*.*

Without the Brjuno condition on subsets, if $z\in\mathbb{C}^{d}$ is such that $z_{n}\to 0$ and $z_{n}\in W(\beta_{1})$ eventually for some $\beta_{1}\in(0,1/d)$ such that $\beta_{1}(l+d-1)>k+1$, we have

[TABLE]

with $1-\frac{\beta_{1}(l+d-1)-1}{k}<0$ and Part (1) through (4) of Proposition 3.2 still follow for these orbits in the same manner (cf. [BRS, Lemma 2.5]). However, in this case we have not been able to determine the containing Fatou components, as both the methods of the next section and of [BRS, Section 5] rely on the Brjuno condition on subsets (cf. remarks in [BRS, Section 1]).

3.1. Global basins are Fatou components

We show that the global basins corresponding to our local basins are Fatou components and conclude the proof of the first part of Theorem 1.

Definition 3.6**.**

Let $F\in\operatorname{Aut}(\mathbb{C}^{d})$ be as in Theorem 1.3. Then for $h=0,\ldots,k-1$, the global basin corresponding to the local basin $B_{h}$ is

[TABLE]

and contains all points $z\in\mathbb{C}^{d}$ such that $F^{n}(z)\in B_{h}$ eventually.

Remark 3.7*.*

The global basins $\Omega_{0},\ldots,\Omega_{k-1}$ are growing unions of preimages of $B_{0},\ldots,B_{k-1}$. As such they are still pairwise disjoint, open, invariant and locally uniformly attracted to [math] under $F$. In particular However, unless $F$ is a global automorphism, they may no longer be connected.

Corollary 3.8**.**

Let $F\in\operatorname{Aut}(\mathbb{C}^{d})$ as in Theorem 1.3 such that $\{\lambda_{1},\ldots,\lambda_{d}\}\backslash\{\lambda_{j}\}$ satisfies the Brjuno condition for each $j=1,\ldots,d$. Then the connected components of $\Omega_{0},\ldots,\Omega_{k-1}$ are Fatou components.

Proof.

Let $h\in\{0,\ldots,k-1\}$. By Remark 3.7, each connected component $C$ of $\Omega_{h}$ is contained in a Fatou component $V$. By normality, for each $z\in V$, we have $z_{n}\to 0$, but by Proposition 3.2 that means $z_{n}$ is contained in $B_{h}$ eventually, i.e. $z\in\Omega_{h}$. Therefore $C\subseteq V\subseteq\Omega_{h}$ and since $V$ is connected, it follows that $V=C$. ∎

The first part of Theorem 1 now follows from Corollary 3.8 and Proposition 3.2.

4. Internal dynamics and geometry

In this section we fix $h\in\{0,\ldots,k-1\}$ and introduce two closely related systems of coordinates compatible with the action of $F$ on the local basin. One allows us to study the behaviour of orbits under $F$, and the other extends to a biholomorphism of the corresponding global basin to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. Note that we do not assume the Brjuno condition on subsets in this section.

4.1. Fatou coordinates

We define special coordinates that codify the dynamics of $F$ on $B_{h}$. The first coordinate $\psi$ is a generalisation of the classical Fatou coordinate in one dimension that was introduced in [BRZ13, Prop. 4.3] and examined more precisely in [BRS, Proposition 3.1 and Lemma 3.3], where the following is shown:

Proposition 4.1**.**

For $F$ and $B_{h}$ as in Theorem 1.3, there exists a holomorphic map $\psi:B_{h}\to\mathbb{C}^{*}$, $\operatorname{Re}\psi>0$ such that

[TABLE]

and a constant $c\in\mathbb{C}$ depending only on $F_{{\rm N}}$ such that

[TABLE]

for $z\in B_{h}$ and $U=(z^{1}\cdots z^{d})^{-k}$.

Moreover, there exists $R_{1}>\max\{R_{0},1\}$, $0<\theta_{1}<\theta_{0}$, and $\beta_{0}<\beta_{1}<1/d$ such that the holomorphic map

[TABLE]

is injective.

The map $\psi$ is obtained as the uniform limit of the sequence $\{\psi_{n}\}_{n\in\mathbb{N}}$ of maps $B_{h}\to\mathbb{C}^{*}$ given by

[TABLE]

where $U_{n}=(z_{n}^{1}\cdots z_{n}^{d})^{-k}$ and $z_{n}=F^{n}(z)$ for $n\in\mathbb{N}$.

Remark 4.2*.*

In particular, $\psi(z)\sim U$ as $|U|\to\infty$ and by Proposition 3.2 (and Remark 3.5), $\psi(z_{n})\sim U_{n}$ as $n\to\infty$ uniformly in $z\in B_{h}$.

[BRS, Proposition 3.4] establishes further local coordinates to cover the remaining dimensions. The following is a slight variation that will simplify the definition of global coordinates in Section 4.3:

Proposition 4.3**.**

Let $F$ and $B_{h}$ be as in Theorem 1.3 and $\psi:B_{h}\to\mathbb{C}^{*}$ as in Proposition 4.1. For $j=2,\ldots,d$, there exists a holomorphic maps $\sigma_{j}:B_{h}\to\mathbb{C}^{*}$ such that

[TABLE]

where the root is well-defined in the main branch since $\operatorname{Re}\psi>0$. Moreover, for every $\alpha\in(1-\beta_{0},k)$ we have

[TABLE]

for $z\in B_{h}$ and $u=z^{1}\cdots z^{d}$.

Remark 4.4*.*

In particular for $j\geq 2$, we have $\sigma_{j}(z_{n})\sim z_{n}^{j}$ as $n\to\infty$ uniformly for $z\in B_{h}$.

Proof.

For $2\leq j\leq d$ we will obtain $\sigma_{j}$ as the limit of the sequence $\{\sigma_{j,n}\}_{n}$ of holomorphic maps $B_{h}\to\mathbb{C}^{*}$ defined for $n\in\mathbb{N}$ by

[TABLE]

where $(z_{n}^{1},\ldots,z_{n}^{d})=F^{n}(z)$ as usual. By Proposition 3.2, we have $z_{n}^{j}=O(n^{-1/kd})$ uniformly for $z\in B_{h}$, so

[TABLE]

uniformly for $z\in B_{h}$. To show convergence, we observe that

[TABLE]

where $R(z_{n})=O(\lVert z_{n}\rVert^{l})=O(u_{n}^{\beta_{0}l})$, since $z_{n}\in B_{h}$. Therefore with (4.5), we obtain

[TABLE]

To estimate the first term on the right hand side, note that by (4.2) we have

[TABLE]

and since $|u_{n}|=O(n^{-1/k})$ it follows

[TABLE]

for any $\alpha\in(1/d,\min\{k,d\})$. Hence, again using $|u_{n}|=O(n^{-1/k})$, (4.6) implies

[TABLE]

Since $2-\alpha/k>1$ and $\beta_{0}l-1/d-\alpha>2k-\alpha>k$, the $O$-terms are summable, and for all $m\geq 0$, we have $u_{n+m}=O(u_{n}),$ so summing up (4.7), we obtain

[TABLE]

Since $\{u_{n}\}_{n}$ converges to [math] uniformly as $n\to\infty$, (4.8) implies that $\{\sigma_{j,n}\}_{n}$ converges uniformly to a holomorphic map $\sigma_{j}:B_{h}\to\mathbb{C}$. For $n=0$ and $m\to\infty$, (4.8) implies

[TABLE]

showing (4.4).

It remains to show that $\sigma_{j}\neq 0$. Since $\sigma_{j,n}\neq 0$ for all $n\in\mathbb{N}$, Hurwitz’s theorem implies that either $\sigma_{j}\equiv 0$ or $\sigma_{j}(z)\neq 0$ for all $z\in B_{h}$. For $r>0$ sufficiently small, we have $(r,\ldots,r)\in B_{h}$ and, by (4.4), we have

[TABLE]

Since $\alpha>(d-1)/d$, this is non-zero for sufficiently small $r>0$, showing that $\sigma_{j}\not\equiv 0$.

Finally, for all $z\in B_{h}$ and $n\in\mathbb{N}$, we have

[TABLE]

proving (4.3). ∎

In the following, we will work in the variables $U=(z^{1}\cdots z^{d})^{-k}$ and $z^{\prime}=(z^{2},\ldots,z^{d})$. Recalling the representation in Remark 1.4 and noting that $u\to u^{-k}$ is injective on $S_{h}(R_{0},\theta_{0})$, the variables $(U,z^{\prime})$ still form a coordinate system on $B_{h}$ in which $B_{h}$ becomes

[TABLE]

The next result, following [BRS, Proposition 3.5], ensures that the maps $\psi,\sigma_{2},\ldots,\sigma_{d}$ still form a coordinate system and their image contains a possibly smaller copy of (4.9).

Proposition 4.5**.**

Let $F$ and $B_{h}$ be as in Theorem 1.3 and $\psi,\sigma_{2},\ldots,\sigma_{j}:B_{h}\to\mathbb{C}^{*}$ as in Propositions 4.1 and 4.3. Then there exist $R_{1}>R_{0}$, $0<\theta_{1}<\theta_{0}$, and $\beta_{0}<\beta_{1}<1/d$ such that the holomorphic map

[TABLE]

is injective. There further exist $R_{2}>0$, $\theta_{2}\in(0,\nicefrac{{\pi}}{{2k}})$, and $\beta_{2}\in(0,1/d)$ such that

[TABLE]

Proof.

Take $R_{1}>\max\{R_{0},1\}$, $0<\theta_{1}<\theta_{0}$, and $\beta_{0}<\beta_{1}<1/d$ from Proposition 4.1. Then for each $n\in\mathbb{N}$, the map

[TABLE]

is injective on $B_{h}(R_{1},\theta_{1},\beta_{1})$. Hence, by Hurwitz’s theorem, the uniform limit $\phi=(\psi,\sigma_{2},\ldots,\sigma_{d})$ of the sequence $\{\phi_{n}\}_{n}$ is either injective or constant on $B_{h}(R_{1},\theta_{1},\beta_{1})$.

As before, for $r>0$ sufficiently small, the point $(r,\ldots,r)$ lies in $B_{h}(R_{1},\theta_{1},\beta_{1})$. We will show that for small values of $r>0$ the Jacobian of $\phi$ at $(r,\ldots,r)$ does not vanish. To simplify calculations, we work in coordinates $(U,z^{\prime})$ as above, so we compute the Jacobian of $\phi:T_{0}:=T(R_{0},\theta_{0},\beta_{0})\to(\mathbb{C}^{*})^{d}$ at

[TABLE]

By Propositions 4.1 and 4.3, we have

[TABLE]

for $z\in B_{h}$.

Observe that since $R_{1}>\max\{R_{0},1\}$, $\theta_{1}<\theta_{0}$, and $\beta_{1}>\beta_{0}$, we have

[TABLE]

so for any $(U^{\prime},z^{\prime})\in T(R_{1},\theta_{1},\beta_{1})$ and $t\in\mathbb{R}$, we have $U_{r}+\delta_{0}e^{it}\in H(R_{0},\theta_{0})$,

[TABLE]

and

[TABLE]

implying $(U+\delta_{0}e^{it},z^{\prime})\in T(R_{0},\theta_{0},\beta_{0})$. In particular for $r>0$ such that $w_{r}\in T(R_{1},\theta_{1},\beta_{1})$, and $t\in\mathbb{R}$ we have

[TABLE]

and for all $j\geq 2$ we have

[TABLE]

where $\delta_{1}=1-R_{1}^{-(1/d-\beta_{0})/k}>0$. Let $h:B_{h}\to\mathbb{C}$ with $h(U,z^{\prime})=O(U^{-\nu})$ for $(U,z^{\prime})\in T_{0}$ for some $\nu>0$. Then there exists $g:B_{h}\to\mathbb{C}$ such that $g(U,z^{\prime})=O(1)$ and $h(U,z^{\prime})=U^{-\nu}g(U,z^{\prime})$ for $(U,z^{\prime})\in T_{0}$. For $w_{r}\in T_{1}$, by (4.11), we then have

[TABLE]

and by (4.12), we have

[TABLE]

Hence, for all $i,j\geq 2$, we obtain

[TABLE]

So for the products in the Jacobian firstly we have

[TABLE]

and secondly for every $\rho\in S_{d}\backslash\{\operatorname{id}\}$, there exists $j\leq d$ such that $\rho(j)\neq j$ and hence $\frac{\partial\phi_{j}}{\partial x^{\rho(j)}}=O(r^{\alpha d-1})$, so we have

[TABLE]

In conclusion the Leibniz formula yields:

[TABLE]

and since $\alpha d>d-1$, this is non-zero for sufficiently small $r>0$, showing that $\phi$ is injective on $B(R_{1},\theta_{1},\beta_{1})$.

Now let $R_{1}^{\prime}>R_{1}$, $0<\theta_{1}^{\prime}<\theta_{1}$, $\beta_{1}<\beta_{1}^{\prime}<1/d$, and $T_{1}^{\prime}:=T(R_{1}^{\prime},\theta_{1}^{\prime},\beta_{1}^{\prime})$ then the closure $\overline{T_{1}^{\prime}}$ is contained in $T_{1}$. To show (4.10), we show that there exist $R_{2}>1$, $\theta_{2}\in(0,\nicefrac{{\pi}}{{2k}})$, and $\beta_{2}\in(0,1/d)$, such that

[TABLE]

Since $\phi$ is an embedding of a neighbourhood of $\overline{T_{1}^{\prime}}$, we have $\partial\phi(T_{1}^{\prime})=\phi(\partial T_{1}^{\prime})$, and, since $T_{2}$ is connected, (4.13) implies $T_{2}\subseteq\phi(T_{1}^{\prime})$.

Fix $0<\theta_{2}<\theta_{1}$ and $\beta_{1}<\beta_{2}<1/d$ and let $w=(U,z^{\prime})\in\partial T^{1}$. We have three cases:

1. * Case* 1.

If $U\in\partial H(R_{1}^{\prime},\theta_{1}^{\prime})$, there exists $C>0$ such that

[TABLE]

for all $|U|>R_{1}+1$ and every $R_{2}>0$, and by (4.2) we have

[TABLE]

so for $|U|$ large enough, we have

[TABLE]

for any $R>0$. 2. * Case* 2.

If $|U|^{(\beta_{1}^{\prime}-1)/k}=|z^{2}\cdots z^{d}|$, then from (4.4) and Remark 4.2 it follows

[TABLE]

so for $|U|$ large enough $|\sigma_{2}(w)\cdots\sigma_{d}(w)|<|\psi(w)|^{(\beta_{2}-1)/k}$. 3. * Case* 3.

If $|z^{j}|=|U|^{-\beta_{1}^{\prime}/k}$, then by (4.4) and Remark 4.2 we have

[TABLE]

so for $|U|$ large enough, $|\sigma_{j}(w)|>|\psi(z)|^{-\beta_{2}/k}$.

In conclusion, there exists $R_{3}>0$ such that $\phi(U,z^{\prime})\notin T(R,\theta_{2},\beta_{2})$ for every $R>0$ and $(U,z^{\prime})\in\partial T_{1}^{\prime}$ such that $|U|>R_{3}$. Since $\operatorname{Re}\psi(z)\approx\operatorname{Re}U\approx|U|$, we can take $R_{2}$ large enough that $\operatorname{Re}\psi(z)<R_{2}$ whenever $|U|\leq R_{3}$, so $\phi(w)\notin T(R_{2},\theta_{2},\beta_{2})$ for all $w\in\partial T_{1}^{\prime}$.

Let again $w_{r}:=(U_{r},z_{r}^{\prime}):=(r^{-kd},r,\ldots,r)$ for $r>0$. As in (4.14), by (4.2) we have

[TABLE]

hence for $r^{-kd}>R_{2}$ large enough,

[TABLE]

and $\psi(w_{r})\in H(R_{2},\theta_{2})$. Again by (4.2) and (4.4), for small $r>0$, we have

[TABLE]

Hence $\phi(w_{r})\in T(R_{2},\theta_{2},\beta_{2})$ for $r>0$ small enough and we have shown (4.13). ∎

4.2. Orbit behaviour

The coordinates from the previous section give us some more precise information about the dynamics in $B_{h}$.

Notation 4.6*.*

For $z=(z^{1},\ldots,z^{d})\in\mathbb{C}^{d}$, denote $|z|:=(|z^{1}|,\ldots,|z^{d}|)$.

Proposition 4.7**.**

Let $F$ and $B_{h}$ be as in Theorem 1.3. Then for $z\in B_{h}$ and $z_{n}=F^{n}(z)$ for $n\in\mathbb{N}$, the limit

[TABLE]

is a unit vector with positive entries and the set of accumulation points of the sequence of directions $\{z_{n}/\lVert z_{n}\rVert_{2}\}_{n}$ is

[TABLE]

In other words, $\{z_{n}\}_{n}$ converges to [math] tangent to the linear cone $\mathbb{R}_{0}^{+}\cdot\omega(z)$.

Furthermore, for any positive unit vector $v\in\mathbb{R}_{+}^{d}$, there exists $z\in B_{h}$ such that $v(z)=v$. Hence the set of all accumulation points of directions of orbits in $B_{h}$ is

[TABLE]

Proof.

Let $\psi$ and $\sigma_{2},\ldots,\sigma_{d}$ be as in Propositions 4.1 and 4.3 and set

[TABLE]

where the root is well-defined, since $\operatorname{Re}\psi>0$ and we always choose its values in the main branch. However, since $u_{n}=z_{n}^{1}\cdots z_{n}^{d}\in S_{h}(R_{0},\theta_{0})$ is near the direction $e^{2\pi ih/k}$ for $z\in B_{h}$, we have $u_{n}=e^{2\pi ih/k}U^{-1/k}$, hence

[TABLE]

as $n\to\infty$. Moreover, the functional equation (4.3) for $\sigma_{j},j\geq 2$ implies the same for $\sigma_{1}$:

[TABLE]

Hence, for $\sigma=(\sigma_{1},\ldots,\sigma_{d})$, we have

[TABLE]

as $n\to+\infty$, where $\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$, so the limit in (4.15) exists and is equal to

[TABLE]

Since $\arg(\sqrt[k]{\psi})+\arg(\sigma_{1})+\cdots+\arg(\sigma_{d})\equiv 2\pi h/k$ by definition, $\sqrt[kd]{\psi}\sigma(z)/\lVert\sqrt[kd]{\psi}\sigma(z)\rVert$ lies in $\omega(z)$ and since (4.16) is invariant under multiplication by $\Lambda$, so does every accumulation point of $\{z_{n}/\lVert z_{n}\rVert_{2}\}_{n}$.

To show that all points in (4.16) occur, note that one-resonance implies that the angles $\arg(\lambda_{2}),\ldots,\arg(\lambda_{d})$ are rationally independent modulo $2\pi$, and this implies, e.g. by [Zeh10, Corollary I.7], that the sequence $\{(\lambda_{2}^{n},\ldots,\lambda_{d}^{n})\}_{n}$ is dense in $(S^{1})^{d-1}$. Hence (4.17) shows that $\{z_{n}/\lVert z\rVert_{2}\}_{n}$ accumulates on the whole set (4.16).

Finally, let $v=(v^{1},\ldots,v^{d})\in\mathbb{R}^{+}$ and $R_{2},\theta_{2},\beta_{2}$ as in Proposition 4.5, so $B_{h}(R_{2},\theta_{2},\beta_{2})\subseteq\sigma(B_{h})$. For $\varepsilon>0$, let v_{\varepsilon}:=\varepsilon e^{2\pi i{\scriptstyle h}/\raisebox{-1.0pt}{{\scriptstyle kd}}}v. Then for $\varepsilon>0$ small enough, we have $\pi(v_{\varepsilon})=\varepsilon^{d}v^{1}\cdots v^{d}e^{2\pi ih/k}\in S_{h}(R_{2},\theta_{2})$ and

[TABLE]

since $d\beta_{2}<1$, and hence $v_{\varepsilon}\in B_{h}(R_{2},\theta_{2},\beta_{2})\subseteq\sigma(B_{h})$, i.e. there exists $z\in B_{h}$ such that $\sigma(z)=v_{\varepsilon}$ and

[TABLE]

Remark 4.8* ($d=2$).*

For $d=2$ and $(z_{0},w_{0})\in B_{h}$, let $m=v(z_{0},w_{0})\in(0,+\infty)$. Then the linear cone $\mathbb{R}_{0}^{+}\cdot\omega(z_{0},w_{0})$ is in fact a real $2$-dimensional linear subspace of $\mathbb{C}^{2}$ given by $\{z=me^{2\pi ih/k}\overline{w}\}$. The complex lines intersecting this subspace are precisely those of the form $\{z=me^{it}w\}$ for $t\in\mathbb{R}$ and all intersections are transversal, so it is not contained in any proper complex subspace of $\mathbb{C}^{2}$.

Recall the representation in Figure 1 of $B_{h}$ in polar decomposition from Remark 1.5. In polar coordinates $(z,w)=(r_{1}e^{is},r_{2}e^{it})$, the linear cone $\mathbb{R}_{+}\cdot\omega(z_{0},w_{0})$ has the form

[TABLE]

Hence Proposition 4.7 translates to the fact that the modulus component converges to [math] tangential to the line $|z|=m|w|$ and the argument component accumulates on the whole central curve $s+t\equiv 2\pi h/k$. Moreover, each value $m\in(0,+\infty)$ occurs, consistent with the fact that the modulus component contains lines with any possible slope $m\in(0,+\infty)$.

Remark 4.9* ($d>2$).*

For $d>2$ and $z\in B_{h}$, the punctured linear cone $\mathbb{R}_{+}\cdot\omega(z)$ still forms a real $d$-dimensional submanifold of $\mathbb{C}^{d}$, but its closure $\mathbb{R}_{0}^{+}\cdot\omega(z)$, has a singularity at [math]. In fact $\mathbb{R}_{+}\cdot\omega(z)$ is not even contained in any proper real subspace of $\mathbb{C}^{d}$.

Proof.

Let $(v^{1},\ldots,v^{d})=v(z)$ and $\zeta$ a primitive $d-1$-st root of unity. Then any real subspace containing $\mathbb{R}_{+}\cdot\omega(z)$ already contains

[TABLE]

and similarly $e_{j}$ and $ie_{j}$ for $j=2,\ldots,d$, hence it has to be $\mathbb{C}^{d}$. ∎

In particular, the above remarks imply:

Corollary 4.10**.**

No orbit of $F$ inside the basins $B_{0},\ldots,B_{k-1}$ as in Theorem 1.3 converges to [math] tangent to a proper complex subspace of $\mathbb{C}^{d}$.

Proof.

Assume $z\in B_{h}$ and $\{z_{n}\}_{n}$ converges to [math] tangent to a complex subspace $V\subseteq\mathbb{C}^{d}$. Then by Proposition 4.7 $V$ has to contain the linear cone $\mathbb{R}^{+}\cdot\omega(z)$. Thus Remarks 4.8 and 4.9 imply that $V=\mathbb{C}^{d}$. ∎

Remark 3 now follows from Proposition 4.7 and Corollary 4.10.

4.3. Geometry of the global basins

By jet-interpolation, we may choose $F$ to be a global automorphism of $\mathbb{C}^{d}$. We then use a variant of the coordinates on each local basin from Section 4.1 that extends to a biholomorphism from the corresponding global basin to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$.

We use the following result from [Wei98] and [For99, Corollary 2.2]:

Theorem 4.11**.**

For every invertible germ of endomorphisms $F_{0}$ of $\mathbb{C}^{d}$ at the origin and every $l\in\mathbb{N}$, there exists an automorphism $F\in\operatorname{Aut}(\mathbb{C}^{d})$ such that $F(z)=F_{0}(z)+O\lparen{{\lVert z\rVert}}^{l}\rparen.$

For $F_{0}=F_{{\rm N}}$ and $l\in\mathbb{N}$ as in Theorem 1.3, this implies that there exist biholomorphisms $F$ of $\mathbb{C}^{d}$ of the form

[TABLE]

with local attracting basins $B_{0},\ldots,B_{k-1}$.

Remark 4.12*.*

For $F\in\operatorname{Aut}(\mathbb{C}^{d})$ of the form (4.19), the global basins $\Omega_{0},\ldots,\Omega_{k-1}$ are growing unions of biholomorphic preimages of $B_{0},\ldots,B_{k-1}$. As such they are still pairwise disjoint and open, invariant and attracted to [math] under $F$, and homotopy equivalent to $(S^{1})^{d-1}$.

To show that these global basins are in fact biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$, we wish to extend the coordinates from the previous section to the global basins via their functional equations (4.1) and (4.3). However, the equation (4.3) involves division by $\psi+1$, which has zeros. In [BRS] this problem is circumvented by restricting to an exhausting sequence of subsets of $\Omega_{h}$ and constructing a fibre bundle biholomorphic to $\Omega_{h}$ with total space $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. We will instead replace $\sigma_{j}$ by a coordinate with a simpler functional equation, that allows for global extension (compare [Rep19]):

Corollary 4.13**.**

Assume the setting of Proposition 4.5. For $2\leq j\leq d$, the map $\tau_{j}=\sqrt[kd]{\psi}\sigma_{j}:B_{h}\to\mathbb{C}^{*}$ is well-defined and satisfies

[TABLE]

Moreover, the map $(\psi,\tau_{2},\ldots,\tau_{d})$ is injective on $B_{h}(R_{1},\theta_{1},\beta_{1})$ and its image contains the set

[TABLE]

Proof.

Fix $2\leq j\leq d$. Since $\operatorname{Re}\psi>0$, the root $\sqrt[kd]{\psi}$ is well-defined. (4.20) follows directly from (4.3):

[TABLE]

Injectivity of $(\psi,\tau_{2},\ldots,\tau_{d})$ and (4.21) follow from Proposition 4.5, since $(\zeta,\xi)\mapsto(\zeta,\sqrt[kd]{\zeta}\xi)$ is well-defined and injective for $\operatorname{Re}\zeta>0$ and $\xi\in\mathbb{C}^{d-1}$, and (4.21) is the image of $T(R_{2},\theta_{2},\beta_{2})$ under that map. ∎

Now if $F$ is an automorphism, this new system of coordinates extends indefinitely:

Proposition 4.14**.**

Let $F$ be an automorphism of the form (4.19), $B_{h}$ as in Theorem 1.3, $\Omega_{h}=\bigcup_{n}F^{-n}(B_{h})$, and $\psi,\tau_{2},\ldots,\tau_{d}:B_{h}\to\mathbb{C}^{*}$ as in Proposition 4.1 and Corollary 4.13. Let $\hat{\psi}:\Omega_{h}\to\mathbb{C}$ and $\hat{\tau}_{2},\ldots,\hat{\tau}_{d}:\Omega_{h}\to\mathbb{C}^{*}$ be given by

[TABLE]

and

[TABLE]

for $z\in F^{-n}(z)$ and $n\in\mathbb{N}$. Then

[TABLE]

is a well-defined biholomorphism. In particular $\Omega_{h}$ is biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$.

Proof.

Let $m>n$ such that $z\in F^{-n}(z)\subseteq F^{-m}(z)$. Then

[TABLE]

and

[TABLE]

so $\hat{\phi}$ is well-defined.

For injectivity, let $z,w\in\Omega_{h}$. Then by Part (3) of Proposition 3.2 (and Remark 3.5) there exists $n\in\mathbb{N}$ such that $F^{n}(z),F^{n}(w)\in B_{h}(R_{1},\theta_{1},\beta_{1})$. Now $\hat{\phi}(z)=\hat{\phi}(w)$ implies

[TABLE]

and by injectivity of $\phi$ on $B_{h}(R_{1},\theta_{1},\beta_{1})$ and of $F$ on $\mathbb{C}^{d}$, we have $F^{n}(z)=F^{n}(w)$ and $z=w$, showing that $\hat{\phi}$ is injective.

To show surjectivity, let $(\zeta,\xi^{\prime})\in\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$. Then for $n\in\mathbb{N}$ large enough, we have $\zeta+n\in H(R_{2},\theta_{2})$,

[TABLE]

since $\beta_{2}<1/d$. Hence by (4.21),

[TABLE]

where $\Lambda^{\prime}:=\operatorname{diag}(\lambda_{2},\cdots,\lambda_{d})$, so there exists $z\in B_{h}$ such that $\hat{\phi}(z)=(\zeta+n,(\Lambda^{\prime})^{n}\xi^{\prime})$ and

[TABLE]

showing surjectivity. ∎

The second part of Theorem 1 now follows from Theorem 4.11 and the following corollary to Proposition 4.14:

Corollary 4.15**.**

Let $F$ be an automorphism of the form (4.19), $B_{h}$ as in Theorem 1.3, and $\Omega_{h}=\bigcup_{n}F^{-n}(B_{h})$. There exists a biholomorphic map $\phi_{h}:\Omega_{h}\to\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ conjugating $F$ to

[TABLE]

Proof.

The biholomorphic map $\hat{\phi}$ from Proposition 4.14 conjugates $F$ to

[TABLE]

where $\Lambda^{\prime}:=\operatorname{diag}(\lambda_{2},\cdots,\lambda_{d})$. The map

[TABLE]

is biholomorphic and well-defined up to choice of a logarithm of the invertible matrix $\Lambda^{\prime}$ and further conjugates (4.23) to (4.22), so $\phi_{h}:=\eta\circ\hat{\phi}$ has the required properties. ∎

5. Periodic cycles

In this section we prove Theorem 4 and Corollary 5 via explicit construction. We first show the existence of “roots up to order $l\in\mathbb{N}$” for one-resonant germs:

Lemma 5.1**.**

Let $d,k\in\mathbb{N}^{*}$ and $F_{0}\in\operatorname{Aut}(\mathbb{C}^{d},0)$ be be one-resonant of index $\alpha$ of the form

[TABLE]

where $c\in\mathbb{C}\backslash\{0\}$ and $\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$. Then for every $p\in\mathbb{N}^{*}$ dividing $k$ and $l\in\mathbb{N}$ there exists a germ $F_{p}\in\operatorname{Aut}(\mathbb{C}^{d},0)$, one resonant of index $p\alpha$, of the form

[TABLE]

where $M_{p}=\operatorname{diag}(\mu_{1},\ldots,\mu_{d})$ is such that $M_{p}^{p}=\Lambda$ and $\mu_{1}\cdots\mu_{d}=\zeta_{p}:=e^{2\pi i/p}$, such that for all germs $F$ such that $F(z)=F_{p}(z)+O(\lVert z\rVert^{l})$, the $p$-th iterate $F^{p}$ has the form

[TABLE]

Proof.

We first determine the iterates of the general germ

[TABLE]

with $a,b\in\mathbb{C}$ and $M$ diagonal such that $M^{k\alpha}=1$. Then for every $m\in\mathbb{N}$, the $m$-th iterate has the form

[TABLE]

and we have

[TABLE]

so

[TABLE]

From this, we obtain and solve recursive expressions for $m\in\mathbb{N}$:

[TABLE]

So in particular

[TABLE]

Choose now $M=M_{p}=\operatorname{diag}(\mu_{1},\ldots,\mu_{d})$ such that $M_{p}^{p}=\Lambda$ and $\mu^{\alpha}=\zeta_{p}$, so $F_{1}$ is one-resonant of index $p\alpha$. For $a=c/p$ and $b=-\frac{p-1}{2}a^{2}(k|\alpha|+1)$, we then have

[TABLE]

Now, by the construction of normal forms for one-resonant germs in [BZ13, Theorem 3.6], for any $l\in\mathbb{N}$, there exists a local holomorphic change of coordinates of the form $\chi(z)=z(1+O(\lVert z\rVert^{3k|\alpha|}))$ such that

[TABLE]

The map $F_{1}$ under this change of coordinates becomes

[TABLE]

and for any $F(z)=F_{p}(z)+O(\lVert z\rVert^{l})$, we have

[TABLE]

Applying Lemma 5.1 to $F_{0}=F_{{\rm N}}$ as in (1.1) and $l>2kd+1$, shows that for every $p\in\mathbb{N}$ dividing $k$, there exists a germ $F_{p}$ of the form

[TABLE]

where $M_{p}=\operatorname{diag}(\mu_{1},\ldots,\mu_{d})$ with $\mu_{1}\cdots\mu_{d}=\zeta_{p}$, such that whenever

[TABLE]

we have $G^{p}(z)=F_{{\rm N}}(z)+O(\lVert z\rVert^{l})$. Again by Theorem 4.11, there exists an Automorphism $F\in\operatorname{Aut}(\mathbb{C}^{d})$ of the form (5.1). In this case, $G^{p}(z)$ is an automorphism of the form (4.19) and has $k$ invariant, non-recurrent, attracting Fatou components $\Omega_{0},\ldots,\Omega_{k-1}$ at [math] each biholomorphic to $\mathbb{C}\times(\mathbb{C}^{*})^{d-1}$ via Proposition 4.14, containing the corresponding local basins $B_{0},\ldots,B_{k-1}$ from Theorem 1.3. Hence, as in dimension $1$, for each $h\in\{0,\ldots,k-1\}$, $\Omega_{h}$ is part of a periodic cycle of Fatou components for $G$ whose period divides $p$.

To show that the period is equal to $p$, note that for $r>0$ sufficiently small $z_{r}:=(r,\ldots,\zeta_{k}^{h}r)\in B_{h}$ for each $h\in\{0,\ldots,k-1\}$. Let $\pi(z)=z^{1}\cdots z^{d}$ for $z=(z^{1},\ldots,z^{d})\in B_{h}$ as usual and $\zeta_{m}:=e^{2\pi i/m}$ for $m\in\mathbb{N}$. Then

[TABLE]

and if $r$ is sufficiently small, we have $\pi(F(z_{r}))\in S_{h+k/p}(R_{0},\theta_{0})$, and hence $F(z_{r})\in B_{h+k/p}$ (indices modulo $k$). This shows that $F$ maps $B_{h}$ to $B_{h+k/p}$ and hence the period of $\Omega_{h}$ is equal to $p$, concluding the proof of Theorem 4.

To derive Corollary 5, take an automorphism $F$ of $\mathbb{C}^{m+1}$ with $k/p$ attracting cycles of period $p$ from Theorem 4 and set

[TABLE]

Then the $w$ component of $\{G^{n}\}_{n\in\mathbb{N}}$ is locally uniformly convergent to [math] on all of $\mathbb{C}^{d-m-1}$, so any subsequence $\{G^{n_{\ell}}\}_{\ell\in\mathbb{N}}$ converges locally uniformly around $(z,w)\in\mathbb{C}^{d}$ if and only if $\{F^{n_{\ell}}\}_{\ell\in\mathbb{N}}$ does so around $z$. Thus $(z,w)$ is in the Fatou set of $G$ if and only if $z$ is in the Fatou set of $F$ and the Fatou components of $G$ are precisely of the form $U\times\mathbb{C}^{d-m-1}$ where $U$ is a Fatou component of $F$. If $U$ is non-recurrent, $p$-periodic and attracting to the origin, then so is $U\times\mathbb{C}^{d-m-1}$.