Rigidity of Julia sets of families of biholomorphic mappings in higher dimension

TL;DR

This paper investigates the rigidity properties of Julia sets for certain automorphisms in higher-dimensional complex spaces, focusing on polynomial shift-like maps and skew products of Hénon maps in , revealing conditions under which Julia sets determine the mappings.

Contribution

It establishes new rigidity results linking Julia sets to the automorphisms in , especially for polynomial shift-like maps and Hénon map skew products.

Findings

Shared Julia sets imply map equivalence in certain classes.

Characterization of Julia set invariance under specific automorphisms.

Rigidity conditions for polynomial shift-like maps and Hénon map skew products.

Abstract

The goal of this article is to study a rigidity property of Julia sets of certain classes of automorphisms in $\mathbb{C}^k$, $k \ge 3.$ First, we study the relation between two polynomial shift-like maps in $\mathbb{C}^k$, $k \ge 3$, that share the same backward and forward Julia sets (or non-escaping sets). Secondly, we study the relationship between any pair of skew products of H\'{e}non maps in $\mathbb{C}^3$ having the same forward and backward Julia sets.

Equations

$$P\circ Q=\sigma \circ P\circ Q$$

$$F\circ H=C\circ H\circ F$$

$$S(z_1,\dots ,z_k)=(z_2,\dots ,z_k,a{z}_1+p(z_{k-\nu +1}))$$

$$V_{R,S}^{+}=\bigcup _{i=k-\nu +1}^k{V}_{R,S}^i\text{ and }{V}_{R,S}^{-}=\bigcup _{i=1}^{k-\nu }{V}_{R,S}^i$$

$$V_{R,S}=\left\{z\in {\mathbb{C}}^k:|z_j|\le R\text{ for }1\le j\le k\right\}$$

$$V_{R,S}^i=\left\{z\in {\mathbb{C}}^k\setminus V_{R,S}:|z_i|>\max \{|z_j|,R\}\text{ for }1\le j\ne i\le k\right\}.$$

$$K_S^{\pm }=\{z\in {\mathbb{C}}^k:\text{the sequence}\left(S^{\pm n}(z)\right)\text{is bounded}\},$$

$$U_S^{\pm }={\mathbb{C}}^k\setminus K_S^{\pm }=\bigcup _{n=0}^{\infty }{S}^{\mp n}(V_{R,S}^{\pm }).$$

$$G_S^{+}(z)=\underset{n\to \infty }{\lim }\frac{1}{d^n}{\log }^{+}\Vert S^{n\nu }(z)\Vert$$

$$G_S^{-}(z)=\underset{n\to \infty }{\lim }\frac{1}{d^n}{\log }^{+}\Vert S^{-n(k-\nu )}(z)\Vert$$

$$G_S^{\pm }(z)=\log |z|+O(1)$$

$$C(z_1,z_2,\dots ,z_k)=({\delta }_{-}{z}_1,\dots ,{\delta }_{-}{z}_{k-\nu },{\delta }_{+}{z}_{k-\nu +1},\dots ,{\delta }_{+}{z}_k)\text{ with }|{\delta }_{\pm }|=1$$

$$\mathcal{T}\circ \mathcal{S}=\mathcal{C}\circ \mathcal{S}\circ \mathcal{T}=\mathcal{S}\circ \mathcal{T}\circ \mathcal{C}$$

$$H(\lambda ,x,y)=(c\lambda ,H_{\lambda }(x,y))$$

$$H_{\lambda }=H_{m,\lambda }\circ H_{(m-1),\lambda }\circ \cdots \circ H_{1,\lambda }$$

$$H_{\lambda }(x,y)=({(H_{\lambda })}_1(x,y),{(H_{\lambda })}_2(x,y))$$

$$x_1^{\lambda }={(H_{\lambda })}_1(x,y)=\sum _{j+k=0}^{d/d_m}{A}_{jk}(\lambda )x^j{y}^k\text{ and }{y}_1^{\lambda }={(H_{\lambda })}_2(x,y)=\sum _{j+k=0}^d{B}_{jk}(\lambda )x^j{y}^k.$$

$${(H_{\lambda })}_2(x,y)=c_H{y}^d+q_{\lambda }(x,y)$$

$$c_H=\prod _{j=1}^m{c_j}^{d_{j+1}\cdots d_m}$$

$$H^{-1}(\lambda ,x,y)=(c^{-1}\lambda ,H_{c^{-1}\lambda }^{-1}(x,y))$$

$$H_{\lambda }^{-1}(x,y)=H_{1,\lambda }^{-1}\circ H_{2,\lambda }^{-1}\circ \cdots \circ H_{m,\lambda }^{-1}(x,y)$$

$${\tilde{x}}_1^{\lambda }={(H_{\lambda }^{-1})}_1(x,y)=\sum _{j+k=0}^d{A}_{jk}^{\prime }(\lambda )x^j{y}^k\text{ and }{\tilde{y}}_1^{\lambda }={(H_{\lambda }^{-1})}_2(x,y)=\sum _{j+k=0}^{d/d_1}{B}_{jk}^{\prime }(\lambda )x^j{y}^k.$$

$${(H_{\lambda }^{-1})}_1(x,y)=c_H^{\prime }{x}^d+q_{\lambda }^{\prime }(x,y)$$

$$c_H^{\prime }=\prod _{j=1}^m{\left(c_j{\delta }_j^{-1}\right)}^{d_{j-1}\cdots d_1}$$

$$\tilde{d}=\mathrm{deg}H=\max \{(j+k)+l_{jk}:0\le (j+k)\le d\}.$$

$$\mathrm{deg}(H^n)=\tilde{d}{d}^{n-1}$$

$$x_n^{\lambda }={(H_{c^{n-1}\lambda }\circ \cdots \circ H_{\lambda })}_1(x,y)\text{ and }{y}_n^{\lambda }={(H_{c^{n-1}\lambda }\circ \cdots \circ H_{\lambda })}_2(x,y)$$

$${\tilde{x}}_n^{\lambda }={\left(H_{c^{-n}\lambda }^{-1}\circ \cdots \circ H_{c^{-1}\lambda }^{-1}\right)}_1(x,y)\text{ and }{\tilde{y}}_n^{\lambda }={\left(H_{c^{-n}\lambda }^{-1}\circ \cdots \circ H_{c^{-1}\lambda }^{-1}\right)}_2(x,y)$$

$$V_R^{+}=\left\{(\lambda ,x,y)\in {\mathbb{C}}^3:|y|>\max \{R,|x|,{|\lambda |}^{\tilde{d}+1}\right\}$$

$$V_R^{-}=\left\{(\lambda ,x,y)\in {\mathbb{C}}^3:|x|>\max \{R,|y|\},|\lambda |<1\right\}.$$

$$V_R^{+}=\left\{(\lambda ,x,y):|y|>\max \{R,|x|,{|\lambda |}^{\tilde{d}+1}\}\right\}$$

Full text

1. Introduction

The fundamental dichotomy of a dynamical system comes from splitting the ambient space into the Fatou set and the Julia set. The Fatou set is an open set where the dynamics is tame and the Julia set is the complement of the Fatou set which supports the wild behavior of the dynamical system. The simplest example of dynamical system which exhibit a rich dynamical behaviour, comes from the class of polynomials in $\mathbb{C}$ of degree greater than or equal to $2$. Being the hub of chaos of the dynamical system, most of the Julia sets of polynomials in $\mathbb{C}$, have very complicated fractal structure. A little perturbation of a polynomial map can change the structure of the corresponding Julia set drastically which gives a hint towards the fact that the Julia sets are quite a rigid object. In fact, a result by Beardon ([2]) validates this anticipation which states that for two polynomials $P$ and $Q$ (of degree greater than or equal to $2$), if $J_{P}=J_{Q}$ where $J_{P}$ and $J_{Q}$ are the Julia sets of $P$ and $Q$ respectively, then

[TABLE]

where $\sigma(z)=az+b$ with $\lvert a\rvert=1$ and $\sigma(J_{P})=J_{P}$. Note that it was known for long time that if two polynomials $P$ and $Q$ commute, then their Julia sets coincide. The results obtained in [1], [3], [15], [12] are also pertinent in this direction.

In dimension $2$, an analogue of this kind of rigidity property of the Julia sets of Hénon maps has recently been established in [8] (Theorem 1.1) which shows that if we start with two Hénon maps $H$ and $F$ for which both forward and backward Julia sets coincide, i.e., $J_{H}^{\pm}=J_{F}^{\pm}$, then

[TABLE]

where $C(x,y)=(\delta_{-}x,\delta_{+}y)$ with $\lvert\delta_{\pm}\rvert=1$ and $C(J_{H}^{\pm})=J_{H}^{\pm}$.

In this note, we study rigidity property of Julia sets of some special classes of biholomorphic mappings in $\mathbb{C}^{k}$, for $k\geq 3$. The first class considered here is the class of shift–like polynomial maps introduced by Bedford and Pambuccian in [4]. Recall that a shift–like map of type $\nu$, where $1\leq\nu\leq k-1$, in $\mathbb{C}^{k}$ is a map of the form:

[TABLE]

with $p$ a polynomial in $\mathbb{C}$ of degree greater than or equal to $2$ and with $0\neq a\in\mathbb{C}$. Note that the class of shift–like polynomial maps is a generalization of Hénon maps for dimensions $k>2$. Further for a polynomial $\nu-$shift $S$, $S^{\nu(k-\nu)}$ or $S^{m}$ is a regular map where $m$ is a multiple of both $\nu$ and $k-\nu$ (See [7] for a detailed proof).

First, let us briefly recall a few terminologies associated to shift–like maps. Suppose $S$ is a $\nu-$shift such that $1\leq\nu\leq k-1.$ Recall from [4] that the filtration corresponding to maps $S$ is as follow:

[TABLE]

where

[TABLE]

and

[TABLE]

We define non-escaping sets

[TABLE]

and the escaping sets

[TABLE]

Further, note that $K_{S}^{\pm}\subset V_{R}\cup V_{R,S}^{\mp}$. The Green functions

[TABLE]

and

[TABLE]

are continuous and plurisubharmonic in $\mathbb{C}^{k}$ which vanish precisely on $K_{S}^{\pm}$. Further,

[TABLE]

in $V_{R,S}^{\pm}$. The rigidity theorem for shift–like polynomial maps in $\mathbb{C}^{k}$, for $k\geq 3$, can now be stated as:

Theorem 1.1**.**

Let $S$ be a polynomial shift–like map of type $\nu$, $1\leq\nu\leq k-1$ in $\mathbb{C}^{k}$, $k\geq 3$. Let $T$ be another shift–like polynomial of degree $d\geq 2$, that preserves $K_{S}^{\pm}$, i.e., $T(K_{S}^{\pm})=K_{S}^{\pm}$, then there exists

[TABLE]

such that

[TABLE]

where $\cal{S}=S^{m}$ and $\cal{T}=T^{m}$ with $m=\text{lcm}(\nu,k-\nu)$.

Though, in spirit the proof of Theorem 1.1 in the present paper is similar to the proof of Theorem 1.1 in [8], they deviate significantly from each other. That the Green functions $G_{S}^{\pm}$ associated to a shift–like map $S$ is not necessarily pluriharmonic in $\mathbb{C}^{k}\setminus K_{S}^{\pm}$, is the main reason for this deviation. For a single Hénon map $H$, the Green functions $G_{H}^{\pm}$ are pluriharmonic in $\mathbb{C}^{2}\setminus K_{H}^{\pm}$ which in turn, gives that the pluricomplex Green functions for the non-escaping sets $K_{H}^{\pm}$ are the Green functions $G_{H}^{\pm}$. Hence, if we start with a pair of Hénon maps for which the non-escaping sets coincide, then their Green functions also coincide. Since it is not clear whether the pluricomplex Green functions for $K_{S}^{\pm}$ are $G_{S}^{\pm}$ or not, starting with two shift–like maps with identical non-escaping sets, it is not possible to conclude directly that they have the same Green functions. However, we succeed to show that the two shift–like maps having identical non–escaping sets indeed have identical Green functions using different techniques from [6] and [10].

Another class of maps which we consider in this note, are the skew products of Hénon maps in $\mathbb{C}^{3}$ of the form:

[TABLE]

for $(\lambda,x,y)\in\mathbb{C}^{3}$ with $0\neq c\in\mathbb{C}$. For each $\lambda\in\mathbb{C}$,

[TABLE]

where $H_{j,\lambda}(x,y)=(y,p_{j,\lambda}(y)-\delta_{j}x)$ with $p_{j,\lambda}$ polynomial of degree $d_{j}\geq 2$ having highest degree coefficient $c_{j}\neq 0$ and $\delta_{j}\neq 0$ for $1\leq j\leq m$. Further, $c_{j}$’s, $\delta_{j}$’s and $d_{j}$’s are independent of $\lambda$. Further, the coefficients of the polynomial $p_{j,\lambda}$ vary continuously with $\lambda$ for $1\leq j\leq m$. This class of maps first appeared in [11] in connection to classification of quadratic polynomial automorphisms in $\mathbb{C}^{3}$ and its dynamics was studied in [9]. To address the rigidity property of the Julia sets of these maps, we first study their dynamics generalizing the techniques developed in [9].

For each $\lambda\in\mathbb{C}$, let

[TABLE]

where the degree of ${(H_{\lambda})}_{1}$ is strictly less than that of ${(H_{\lambda})}_{2}$ and in fact, $\deg{(H_{\lambda})}_{1}$ is ${d}/{d_{m}}$ when regarded as a polynomial in $x$ and $y$. Further, let

[TABLE]

In particular,

[TABLE]

where

[TABLE]

with the convention that $d_{j+1}\cdots d_{m}=1$ when $j=m$, $d=d_{m}\cdots d_{1}$ and $q_{\lambda}$ a polynomial in $x$ and $y$ of degree strictly less than $d$, for each $\lambda\in\mathbb{C}$. Now note that

[TABLE]

and

[TABLE]

for each $\lambda\in\mathbb{C}$ and for all $(x,y)\in\mathbb{C}^{2}$. Let

[TABLE]

In particular,

[TABLE]

where

[TABLE]

with the convention that $d_{j-1}\cdots d_{1}=1$ when $j=1$ and $q_{\lambda}^{\prime}$ a polynomial in $x$ and $y$ with degree strictly less than $d$. Further, the maps $\lambda\mapsto A_{jk}(\lambda)$, $\lambda\mapsto B_{jk}(\lambda)$, $\lambda\mapsto A_{jk}^{\prime}(\lambda)$ and $\lambda\mapsto B_{jk}^{\prime}(\lambda)$ are assumed to be continuous in $\mathbb{C}$.

Let $\deg(B_{jk})=l_{jk}$ for $0\leq j+k\leq d$. Define

[TABLE]

Consequently,

[TABLE]

for all $n\geq 1$. Further, we fix the following notation:

[TABLE]

for all $n\geq 1$ and for each $\lambda\in\mathbb{C}$. Similarly, fix

[TABLE]

for all $n\geq 1$ and for each $\lambda\in\mathbb{C}$.

In Section 3, we study the dynamics of the map of the form (1.2). Depending on modulus of $c$, we choose the sets $V_{R}^{+}$ and $V_{R}^{-}$ for sufficiently large $R$. For $\lvert c\rvert>1$, we define

[TABLE]

For $\lvert c\rvert<1$, the role of $V_{R}^{+}$ and $V_{R}^{-}$ gets interchanged. For $\lvert c\rvert=1$, set

[TABLE]

The sets $V_{R}^{\pm}$ help to localize the dynamics of the map $H$. Define

[TABLE]

Further, define the Julia sets (forward and backward) $J_{H}^{\pm}=\partial K_{H}^{\pm}$. Clearly the sets $K_{H}^{\pm}$ and $J_{H}^{\pm}$ are invariant under the map $H$. For each $n\geq 1$, define

[TABLE]

for $(\lambda,x,y)\in\mathbb{C}^{3}$. The sequence of functions $G_{n,H}^{\pm}$ converge to the functions $G_{H}^{\pm}$ uniformly on compacts. The function $G_{H}^{\pm}$ is plurisubharmonic in $\mathbb{C}^{3}$ and pluriharmonic in $\mathbb{C}^{3}\setminus K_{H}^{\pm}$ satisfying the natural functorial property. Further, $G_{H}^{\pm}>0$ in $\mathbb{C}^{3}\setminus K_{H}^{\pm}$ and they vanish precisely on $K_{H}^{\pm}$. Properties of Green functions $G_{H}^{\pm}$ has been recorded in Theorem 3.4.

As in the case of single Hénon map, the sets $K_{H}^{\pm}$ are not necessarily the collection of the points in $\mathbb{C}^{3}$ having bounded orbits. For example, in case $\lvert c\rvert>1$, for $(\lambda,x,y)\in K_{H}^{+}$ with $\lambda\neq 0$, $\lvert H^{n}(\lambda,x,y)\rvert\rightarrow\infty$ as $n\rightarrow\infty$. If $K_{H}^{+}\subseteq\{\lambda=0\}$, then since $G_{H}^{+}$ is continuous in the whole $\mathbb{C}^{3}$ and $G_{H}^{+}$ is pluriharmonic away from $K_{H}^{+}$ , by removable singularity theorem $G_{H}^{+}$ can be extended as pluriharmonic function in whole $\mathbb{C}^{3}$. Now $G_{H}^{+}>0$ in $U_{H}^{+}$. Thus $G_{H}^{+}$ is identically constant in $\mathbb{C}^{3}$. This shows that, in this case $K_{H}^{+}$ always contain a point in $\mathbb{C}^{3}$ with unbounded orbit. In Lemmas 3.2 and 3.6, we give an estimate of the growth rate of a point in $K_{H}^{\pm}$.

We now state the rigidity theorem for skew products of Hénon maps fibered over non-compact parameter space $\mathbb{C}$.

Theorem 1.2**.**

Let $H$ and $F$ be two skew products of Hénon maps in $\mathbb{C}^{3}$. Further, assume that Julia sets of $H$ and $F$ are the same, i.e., $J_{H}^{\pm}=J_{F}^{\pm}$. Then,

[TABLE]

for some $\gamma:(\lambda,x,y)=(\delta\lambda,\delta_{+}x,\delta_{-}y)$ with $0\neq\delta\in\mathbb{C}$ and $\lvert\delta_{\pm}\rvert=1$.

In the same spirit as above, we can give a rigidity theorem for skew-products of Hénon maps fibered over a compact metric space. We conclude this note after discussing the proof of this theorem.

2. Rigidity of Julia sets of shift–like maps

In this section, we first prove a uniqueness result for locally bounded, plurisubharmonic functions having at most logarithmic growth. It is essentially a modification to Theorem 1 from [6]. Let

[TABLE]

Lemma 2.1**.**

Let $u,v\in\mathcal{L}$ such that

[TABLE]

for some $1\leq p\leq(k-1)$, then $u$ and $v$ differ by a constant in $\mathbb{C}^{k}$.

Proof.

Let

[TABLE]

and let $L$ be any strongly plurisubharmonic function in $\mathbb{C}^{k}$. We will show that

[TABLE]

which in turn gives that $d\rho=0$ almost everywhere and hence $\rho$ is a constant.

We prove by induction that for $q=0,\ldots,(k-p-1)$

[TABLE]

on $\mathbb{C}^{k}$ where $i+j=k-q-p-1$.

We prove (2.2) for $q=0$ first i.e., we prove that

[TABLE]

for $i+j=k-p-1$. Let

[TABLE]

Using (2.1), we have that

[TABLE]

Let

[TABLE]

then

[TABLE]

is an exact positive $(k,k)-$current and thus $dT=0$ which in turn gives (2.3).

Suppose inductively, we have

[TABLE]

with $i+j=k-p-q-1$. We will be done if we prove that

[TABLE]

with $i+j=k-p-q-2$. Using Schwarz inequality (2.5) gives that

[TABLE]

and

[TABLE]

Let

[TABLE]

with $i+j=k-p-q-2$. Let

[TABLE]

Note that by using (2.7), it follows that

[TABLE]

Hence,

[TABLE]

As before, $dU$ is an exact, positive, $(k,k)$-current and thus

[TABLE]

This proves (2.6). Thus we get that $u$ and $v$ differ by a constant in $\mathbb{C}^{k}$. ∎

2.1. Proof of Theorem 1.1

Recall that a shift–like map $S$ of type $\nu$ ( $1\leq\nu\leq k-1$) is a map of the form:

[TABLE]

where $a\in\mathbb{C}^{*}$ and $p$ is a polynomial of degree $d\geq 2.$ From Proposition 2.4 in [7] the $\nu(k-\nu)-$th iterate of a shift–like map $S$ is regular. The positive and negative Green functions of $S$ are defined as:

[TABLE]

Further from Theorem 8.7 in [10],

[TABLE]

are unique currents of mass $1$ supported in $K_{S}^{+}$ and $K_{S}^{-}$ respectively.

*Step 1: * $T$ is a $\nu-$shift. Also $G_{S}^{\pm}=G_{T}^{\pm}$ and $K_{S}^{\pm}=K_{T}^{\pm}.$

Let $T$ be a $\nu_{1}-$shift such that $1\leq\nu_{1}\neq\nu\leq k-1.$ Recall from [4], the filtration properties of the maps $S$ and $T$. For $R>0$ we define the sets

[TABLE]

and

[TABLE]

Also,

[TABLE]

for a sufficiently large $R>0.$

Let $\nu_{1}<\nu.$ Then $V_{R,T}^{+}\subset V_{R,S}^{+}$ and $V_{R,S}^{-}\subset V_{R,T}^{-}.$ Pick a $z\in K_{S}^{+}.$ By assumption

[TABLE]

i.e., $z\in K_{T}^{+}.$ Thus $K_{S}^{+}\subset K_{T}^{+}.$

Let $\overline{K_{T}^{\pm}}=K_{T}^{\pm}\cup I^{\pm}_{T}\subset\mathbb{P}^{k}$ and $\overline{K_{S}^{\pm}}=K_{S}^{\pm}\cup I^{\pm}_{S}\subset\mathbb{P}^{k}$ where $I^{\pm}_{S}$ and $I^{\pm}_{T}$ are indeterminacy sets of $S$ and $T$ respectively. Since $\nu_{1}<\nu$, by Proposition 2.1 in [7] it follows that $I_{S}^{+}\subset I_{T}^{+}$ and $I_{T}^{-}\subset I_{S}^{-}.$ Now from Theorem 8.7 in [10] $\mu_{T}^{+}$ is the unique $(\nu_{1},\nu_{1})-$closed current supported in $\overline{K_{T}^{+}}$ and $\mu_{S}^{+}$ is the unique $(\nu,\nu)-$closed current of mass 1 supported in $\overline{K_{S}^{+}}.$ Further from Corollary 8.5 in [10] $\tilde{\mu}_{S}=(\frac{1}{2\pi}dd^{c}G_{S}^{+})^{\nu_{1}}$ is a $(\nu_{1},\nu_{1})-$closed current of mass $1$ supported in $\overline{K_{S}^{+}}$, hence in $\overline{K_{T}^{+}}.$ Since $(\nu_{1},\nu_{1})-$closed current of mass 1 supported in $\overline{K_{T}^{+}}$ is unique,

[TABLE]

Applying Lemma 2.1, we have that $G_{S}^{+}=G_{T}^{+}$ which in turn gives $K_{S}^{+}=K_{T}^{+}.$ As $K_{T}^{+}\subset V_{R}\cup V_{R,S}^{-}$ it follows that $K_{T}^{+}\cap V_{R}^{k-\nu_{1}}=\emptyset.$ But from Proposition 2.1 in [7] and Proposition 8.3 in [10], it follows that

[TABLE]

Hence $\nu_{1}\not\leq\nu.$ A similar argument applied for $T^{-1}$ on $K_{S}^{-}$ and $K_{T}^{-}$ gives $\nu\not\leq\nu_{1}.$ Thus $\nu_{1}=\nu.$

Assuming $\nu_{1}=\nu$, exactly the same arguments as above gives $K_{S}^{+}\subset K_{T}^{+}$ and $K_{S}^{-}\subset K_{T}^{-}.$ Since $\mu_{T}^{\pm}$ and $\mu_{S}^{\pm}$ are unique currents of mass 1 supported in $K_{T}^{\pm}$ and $K_{S}^{\pm}$ respectively,

[TABLE]

i.e.,

[TABLE]

Hence by Lemma 2.1, $G_{S}^{\pm}=G_{T}^{\pm}$ and $K_{S}^{\pm}=K_{T}^{\pm}.$

Similar to the form of $S$ in (2.8) let the form of $T$ be

[TABLE]

where $b\in\mathbb{C}^{*}$ and $q$ a polynomial of degree $d_{q}\geq 2.$ Let

[TABLE]

For $\epsilon>0$ we define the modified sectors $V_{i}^{R}(\epsilon)$, $1\leq i\leq k$ as:

[TABLE]

*Step 2: *There exist modified sectors such that they are invariant under appropriate iterates of $S$ and $T$ or $S^{-1}$ and $T^{-1}.$

Lemma 2.2**.**

Corresponding to the shift–like maps $S$ and $T$ (of type $\nu$) there exist $R_{0}>0$ and $\epsilon_{0}>0$ ($\epsilon_{0}\ll R_{0}$) such that

• (i)

For every $k-\nu+1\leq i\leq k$,

[TABLE]

whenever $R>R_{0}.$

• (ii)

For every $1\leq j\leq k-\nu$,

[TABLE]

whenever $R>R_{0}.$

Proof.

Note that

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

*Claim: * There exists $R_{0}\gg 1$ and $\epsilon_{0}>0$ (sufficiently smaller than $R_{0}$) such that if $z\in D(0;R)$ and $w\in D(0;R+\epsilon_{0})^{c}$ for $R\geq R_{0}$ then

[TABLE]

Recall that $p(z)=c_{d_{p}}z^{d_{p}}+\cdots+c_{0}$ and $q(z)=c^{\prime}_{d_{q}}z^{d_{q}}+\cdots+c^{\prime}_{0}.$ For a given $\epsilon>0$, there exists $R(\epsilon)\gg 1$ such that

[TABLE]

and

[TABLE]

whenever $|z|\geq R(\epsilon).$ Further, we modify $R_{0}>R(\epsilon)$ and choose $\epsilon_{0}$ (sufficiently large) such that if $z\in D(0;R)$ and $w\in D(0;R+\epsilon_{0})^{c}$ where $R\geq R_{0}$, then

[TABLE]

where $d=\min(d_{p},d_{q})$ and $M=\max\{|a|,|b|,1\}.$ Note that

[TABLE]

Further we modify the choice of $R_{0}$, such that $2\epsilon R_{0}^{d}>M\epsilon_{0}.$ Then for $z\in D(0;R)\setminus D(0;R_{0})$ and $w\in D(0;R+\epsilon_{0})^{c}$, where $R>R_{0}$, it follows that

[TABLE]

and

[TABLE]

Now by maximum modulus principle, for each $z\in D(0;R)$ and for each $w\in D(0;R+\epsilon_{0})^{c}$, where $R\geq R_{0}$, we have

[TABLE]

and hence the claim follows.

Now let $z\in V_{i_{0}}^{R}(\epsilon_{0})$ where $k-\nu+1\leq i_{0}\leq k$ with $R>R_{0}.$ Let $|z_{i_{0}}|=\tilde{R}+\epsilon_{0}$, then clearly $|z_{i}|\leq\tilde{R}$ for every $1\leq i\neq i_{0}\leq k$ where $\tilde{R}>R_{0}.$ Observe that

[TABLE]

for $k-\nu+1\leq i\leq k.$

*Claim: *For every $z\in V_{i_{0}}^{R}(\epsilon_{0})$, $|\pi_{i_{0}}S^{\nu}(z)|>|\pi_{i}S^{\nu}(z)|+\epsilon_{0}$, for every $1\leq i\leq k$ and $i\neq i_{0}$ where $R>R_{0}.$

By (2.9) and by definition of $M$ it follows that if $k-\nu+1\leq i\leq k$ and $i\neq i_{0},$ then

[TABLE]

But

[TABLE]

Hence

[TABLE]

for $k-\nu+1\leq i\leq k.$ Thus the claim follows.

Now $R_{0}$ is so chosen, such that $S(V_{R}^{+})\subset V_{R}^{+}$ for every $R>R_{0}$, i.e., S^{\nu}\big{(}V_{i}^{R}(\epsilon_{0})\big{)}\subset V_{i}^{R}(\epsilon_{0}) for $k-\nu+1\leq i\leq k.$ A similar argument will work for $T$ and this completes the proof of part (i).

Now let $z\in V_{i_{0}}^{R}(\epsilon_{0})$ and $1\leq j_{0}\leq k-\nu$ where $R>R_{0}.$ Let $|z_{j_{0}}|=\tilde{R}+\epsilon_{0}$, then clearly $|z_{i}|\leq\tilde{R}$ for every $1\leq i\neq j_{0}\leq k$ and $\tilde{R}>R_{0}.$ Observe that

[TABLE]

for $1\leq j\leq k-\nu.$

Claim: $|\pi_{j_{0}}S^{-(k-\nu)}(z)|>|\pi_{j}S^{-(k-\nu)}(z)|+\epsilon_{0}$ for every $1\leq j\leq k-\nu$ and $j\neq j_{0}.$

By (2.9) and definition of $M$ it follows that if $k-\nu+1\leq j\leq k$ and $j\neq j_{0}$

[TABLE]

But

[TABLE]

Hence

[TABLE]

for $k-\nu+1\leq j\leq k.$ This proves the claim.

Now $R_{0}$ is so chosen, such that $S^{-1}(V_{R}^{-})\subset V_{R}^{-}$ for every $R>R_{0}$, i.e., S^{-(k-\nu)}\big{(}V_{j}^{R}(\epsilon_{0})\big{)}\subset V_{j}^{R}(\epsilon_{0}) for $1\leq j\leq k-\nu.$ A similar argument will work for $T$ and this completes the proof of part (ii). ∎

*Step 3: * Note that Lemma 2.2 assures that appropriate modified sectors are invariant under $S^{\nu}$ or $S^{-(k-\nu)}$. In this step we construct Böttcher coordinates in each of this sectors.

Proposition 2.3**.**

Let $S$ be a polynomial shift–like map of type $\nu.$ Then there exist $R_{S}>0$, $\epsilon_{S}>0$ ($\epsilon_{S}\ll R_{S}$) and holomorphic functions $\phi_{S,i}^{+}$ on $V_{i}^{R_{S}}(\epsilon_{S})$, $k-\nu+1\leq i\leq k$ and $\phi_{S,j}^{-}$ on $V_{j}^{R_{S}}(\epsilon_{S})$, $1\leq j\leq k-\nu$ such that

[TABLE]

Also $\phi_{S,i}^{+}(z)\sim z_{i}$ and $\phi_{S,j}^{-}(z)\sim z_{j}$ for $z\to\infty.$

Proof.

From Lemma 2.2, there exists $R_{0}$ and $\epsilon_{S}$ such that for $1\leq i\leq k$ and $1\leq j\leq k-\nu$

[TABLE]

for every $R>R_{0}.$ Let $z_{i,n}=\pi_{i}\circ S^{\nu n}(z)$ for every $k-\nu+1\leq i\leq k$ and $d=d_{p}$.

*Claim: *The telescoping product

[TABLE]

converges in $V_{i}^{R}(\epsilon_{S})$ for $R$ sufficiently large whenever $k-\nu+1\leq i\leq k$.

Note that

[TABLE]

Since $p(z)=c_{d}z^{d}+\ldots+c_{0}$, (2.10) can be written as

[TABLE]

Now if $R$ is chosen sufficiently large and $z\in V_{i}^{R}(\epsilon_{S})$

[TABLE]

We take an appropriate branch of logarithm such that $\text{Log}\;c_{d}$, $\text{Log}\;a^{-1}c_{d}$ and $\text{Log}\;z$ are well defined for every $z\in\mathbb{D}(1;1)$ (the ball of radius 1 at 1). Hence, the convergence of the telescoping product

[TABLE]

is equivalent to convergence of the series

[TABLE]

Note that (2.12) converges for every $z\in V_{i}^{R}(\epsilon_{s})$, i.e., the function

[TABLE]

is well defined in $V_{i}^{R}(\epsilon_{S}).$ Note that there exists $M>0$ such that if $R>0$ is sufficiently large

[TABLE]

Hence from (2.11) and (2.13) it follows that

[TABLE]

Also

[TABLE]

Thus the proof follows.

A similar argument gives that there exist holomorphic functions $\phi_{j,S}^{-}$ on $V_{j}^{R}(\epsilon_{s})$ for $1\leq j\leq k-\nu$ such that

[TABLE]

and

[TABLE]

∎

Hence by Lemma 2.2 and Proposition 2.3 it is possible to choose $R_{1}$ sufficiently large such that the following are true:

• (i)

There exist holomorphic functions $\phi_{S,i}^{+}$ on $V_{i}^{R_{1}}(\epsilon_{0})$, $k-\nu+1\leq i\leq k$ and $\phi_{S,j}^{-}$ on $V_{j}^{R_{1}}(\epsilon_{0})$, $1\leq j\leq k-\nu$ such that

[TABLE]

Also $\phi_{S,i}^{+}(z)\sim z_{i}$ and $\phi_{S,j}^{-}(z)\sim z_{j}$ for $z\to\infty.$

• (ii)

There exist holomorphic functions $\phi_{T,i}^{+}$ on $V_{i}^{R_{1}}(\epsilon_{1})$, $k-\nu+1\leq i\leq k$ and $\phi_{T,j}^{-}$ on $V_{j}^{R_{1}}(\epsilon_{0})$, $1\leq j\leq k-\nu$ such that

[TABLE]

Also $\phi_{T,i}^{+}(z)\sim z_{i}$ and $\phi_{T,j}^{-}(z)\sim z_{j}$ for $z\to\infty.$

*Step 4: *The Böttcher coordinates of $S^{\nu}$, $S^{-(k-\nu)}$ and $T^{\nu}$, $T^{-(k-\nu)}$ are equal in appropriate modified sectors.

Lemma 2.4**.**

For $k-\nu+1\leq i\leq k$, on $V_{i}^{R_{1}}(\epsilon_{0})$, $\phi_{S,i}^{+}=\phi_{T,i}^{+}$ and for $1\leq j\leq k-\nu$, on $V_{j}^{R_{1}}(\epsilon_{0})$, $\phi_{S,j}^{-}=\phi_{T,j}^{-}.$

Proof.

Let $R_{0}$ and $\epsilon_{0}$ be as chosen in Lemma 2.2. Then for every $R>R_{0}$:

• (i)

For $k-\nu+1\leq i\leq k$ and $z\in V_{i}^{R}(\epsilon_{0})$

[TABLE]

• (ii)

For $1\leq j\leq k-\nu$ and $z\in V_{j}^{R}(\epsilon_{0})$

[TABLE]

Now from the proof of Proposition 2.3, for $k-\nu+1\leq i\leq k$ and $z\in V_{i}^{R_{1}}(\epsilon_{0})$

[TABLE]

and for $1\leq j\leq k-\nu$ and $z\in V_{j}^{R_{1}}(\epsilon_{0})$

[TABLE]

Hence

[TABLE]

and

[TABLE]

Since $G_{S}^{+}=G_{T}^{+}$ (by Step 1), for $k-\nu+1\leq i\leq k$ and for $z\in V_{i}^{R_{1}}(\epsilon_{0})$

[TABLE]

Now by Lemma 2.3, both $\phi_{S,i}^{+}(z),\phi_{T,i}^{+}(z)$ asymptotic to $z_{i}$ as $z\to\infty$ in $V_{i}^{R_{1}}(\epsilon_{0})$, i.e.,

[TABLE]

A similar argument gives that on $V_{j}^{R_{1}}(\epsilon_{1})$, $1\leq j\leq k-\nu$

[TABLE]

∎

From now onwards we will use $\phi_{i}^{+}$, to denote $\phi_{S,i}^{+}=\phi_{T,i}^{+}$ on $V_{i}^{R_{1}}(\epsilon_{0})$, for $k-\nu+1\leq i\leq k$ and $\phi_{j}^{-}$, to denote $\phi_{S,j}^{-}=\phi_{T,j}^{-}$ on $V_{j}^{R_{1}}(\epsilon_{0})$, for $1\leq j\leq k-\nu.$

*Step 5: * The coordinates $\mathcal{S}\circ\mathcal{T}(z)$ is related to the coordinates of $\mathcal{T}\circ\mathcal{S}(z)$ where $\cal{S}=S^{m}$, $\cal{T}=T^{m}$ and $m=lcm(\nu,k-\nu).$

Lemma 2.5**.**

1. (i)

For every $z\in\mathbb{C}^{k}$ and for $k-\nu+1\leq i\leq k$ there exists $\delta_{+}$ with $|\delta_{+}|=1$ such that

[TABLE] 2. (ii)

For every $z\in\mathbb{C}^{k}$ and for $1\leq j\leq k-\nu$ there exist $\delta^{j}_{-}$’s with $|\delta^{j}_{-}|=1$ such that

[TABLE]

Proof.

Let $m=m_{1}\nu=m_{2}(k-\nu).$ From Lemma 2.3, on $V_{i}^{R_{1}}(\epsilon_{0})$, $k-\nu+1\leq i\leq k$, we have

[TABLE]

Further from (2.15) there exists $\delta_{1}\in\mathbb{C}$ such that $|\delta_{1}|=1$ and

[TABLE]

From (2.19) thus on $V_{i}^{R_{1}}(\epsilon_{0})$, $k-\nu+1\leq i\leq k$, we have the following:

[TABLE]

Hence from (2.20) on $V_{i}^{R_{1}}(\epsilon_{0})$, $k-\nu+1\leq i\leq k$ we have

[TABLE]

where $\delta_{+}=\delta_{1}^{m_{1}}.$ From Lemma 2.3, if $z\to\infty$ on $V_{i}^{R_{1}}(\epsilon_{0})$,

[TABLE]

and

[TABLE]

So from (2.21),

[TABLE]

Fix $z_{0}\in\mathbb{C}^{k-1}$ and consider the line $L_{z_{0}}^{i}=\{(\xi,z_{0})\in\mathbb{C}^{k}:\pi_{i}(\xi,z_{0})=\xi\}.$ Then $(\xi,z_{0})\in V_{i}^{R_{1}}(\epsilon_{0})$ for $|\xi|$ sufficiently large. Also $(\xi,z_{0})\to\infty$ if $\xi\to\infty.$ Thus from (2.22) it follows that

[TABLE]

as $\xi\to\infty.$ Since (2.23) is a polynomial in $\xi$, it has to be identically the zero polynomial, i.e.,

[TABLE]

Now (2.24) is true for any $z_{0}\in\mathbb{C}^{k-1}$, hence (2.17) is true. This completes the proof of (i).

From Proposition 2.3, on $V_{j}^{R_{1}}(\epsilon_{0})$, $1\leq j\leq k-\nu$ we have

[TABLE]

Therefore for z\in\cal{S}^{-1}\big{(}V_{j}^{R_{1}}(\epsilon_{0})\big{)}, $1\leq j\leq k-\nu$ we have

[TABLE]

and z\in\cal{T}^{-1}\big{(}V_{j}^{R_{1}}(\epsilon_{0})\big{)}, $1\leq i\leq k-\nu$

[TABLE]

Let U_{j}^{1}=(\cal{S}\circ\cal{T})^{-1}\big{(}V_{j}^{R_{1}}(\epsilon_{0})\big{)}, and U_{j}^{2}=(\cal{T}\circ\cal{S})^{-1}\big{(}V_{j}^{R_{1}}(\epsilon_{0})\big{)}, $1\leq i\leq k-\nu.$ Note that $U_{j}^{1}\cap U_{j}^{2}\neq\emptyset$ as both $U_{j}^{1}$ and $U_{j}^{2}$ are neighbourhood of the point

[TABLE]

Let $\cal{U}_{j}=U_{j}^{1}\cap U_{j}^{2}$ for $1\leq j\leq k-\nu.$ Thus for $z\in\cal{U}_{j}$

[TABLE]

and

[TABLE]

Further from (2.16) there exists $\delta_{2}\in\mathbb{C}$ such that $|\delta_{2}|=1$ and

[TABLE]

Combining (2.26)–(2.28), for $z\in\cal{U}_{j}$, we have that

[TABLE]

For every $1\leq j\leq k-\nu$ we take appropriate root $\delta_{2}^{m_{2}}$, say $\delta_{-}^{j}$ (note that $|\delta_{-}^{j}|=1$) on $\cal{U}_{j}$ we have that

[TABLE]

Fix a $j$, $1\leq j\leq k-\nu$ and $c=(c_{1},\ldots,c_{k-1})\in\mathbb{C}^{k-1}$ . Further for $\xi\in\mathbb{C}$, consider the points in the projective space

[TABLE]

Let $\xi_{n}\in\mathbb{C}$ such that $|\xi_{n}|\to\infty.$ Then

[TABLE]

i.e., $x_{n}=(c_{1},\ldots,c_{j-1},\xi_{n},c_{j}\ldots,c_{k-1})\in\cal{U}_{j}$ for $n$ sufficiently large. Note

[TABLE]

and

[TABLE]

are polynomials in $\mathbb{C}.$ Thus $P_{1}(\xi_{n})\to\infty$, $P_{2}(\xi_{n})\to\infty$ as $n\to\infty$, i.e.,

[TABLE]

and

[TABLE]

as $n\to\infty.$ Therefore from Lemma 2.3 we have that

[TABLE]

and

[TABLE]

Now from (2.30), it follows that

[TABLE]

as $n\to\infty.$ Since $P_{2}-\delta_{-}^{j}P_{1}$ is polynomial in one variable, the above is not possible unless it is the zero polynomial. Hence for every $\xi\in\mathbb{C}$

[TABLE]

Now the above observation is true for any $c\in\mathbb{C}^{k-1}$, thus (2.18) holds. This completes the proof of (ii). ∎

Thus there exists a linear map $C_{1}(z_{1},\ldots,z_{k})=(\tilde{\delta_{1}}z_{1},\ldots,\tilde{\delta_{k}}z_{k})$ such that

[TABLE]

where $\tilde{\delta_{j}}=\delta_{-}^{j}$ for $1\leq j\leq k-\nu$ and $\tilde{\delta_{i}}=\delta_{+}$ for $k-\nu+1\leq i\leq k.$

*Step 6: * The coordinates of $\cal{T}^{-1}\circ\cal{S}^{-1}$ and $\cal{S}^{-1}\circ\cal{T}^{-1}$ are also related in a similar way.

Lemma 2.6**.**

There exists a linear map

[TABLE]

where $\delta^{\prime}_{j}=\delta^{\prime}_{-}$ for $1\leq j\leq k-\nu$ and $\delta^{\prime}_{i}={\delta^{\prime}_{+}}^{i}$ for $k-\nu+1\leq i\leq k$ such that

[TABLE]

Proof.

The proof is similar to Lemma 2.5 if the role of $\cal{T}$ and $\cal{S}$ are interchanged with the role of $\cal{T}^{-1}$ and $\cal{S}^{-1}$ appropriately. ∎

*Step 7: *Last step is to prove that the maps $C_{1}$ and $C_{2}$ are equal.

From (2.31) and (2.32) it follows that

[TABLE]

Let $D_{1}$, $D_{2}$ be diagonal matrices defined as follows:

[TABLE]

Now by applying chain rule to (2.33) it follows that

[TABLE]

where $A=D(\cal{S}\circ\cal{T})(0)$. Note that $A$ is invertible and $A$, $D_{1}$, $D_{2}$ has the following forms:

[TABLE]

where $\cal{D}_{1}=\text{Diag}({\delta_{-}^{1}},\ldots,{\delta_{-}^{k-\nu}})$ and $\cal{D}_{2}=\text{Diag}({\delta^{\prime}}_{+}^{k-\nu+1},\ldots,{\delta^{\prime}}_{+}^{k}).$ Thus (2.34) simplifies as:

[TABLE]

Case 1: Suppose $\nu\neq k-\nu$, i.e., either $\nu<k-\nu$ or $k-\nu<\nu.$

Assume $k-\nu<\nu.$ Since $A$ is invertible, the possibilities are

• (i)

$\cal{A}$ and $\cal{D}$ should be invertible.

• (ii)

If $\cal{A}$ or $\cal{D}$ is not invertible, then $\text{Rank}(\cal{B})=k-\nu.$

*Subcase (i): * If $\cal{A}$ and $\cal{D}$ are invertible, then $\cal{D}_{1}=\delta^{\prime}_{-}\text{Id}$ and $\cal{D}_{2}=\delta_{+}\text{Id}$, i.e., $\delta_{-}^{i}=\delta^{\prime}_{-}$ for $1\leq i\leq k-\nu$ and ${\delta^{\prime}}_{+}^{i}=\delta_{+}$ for $k-\nu+1\leq i\leq k.$ This proves that for $|\delta_{\pm}|=1$,

[TABLE]

*Subcase (ii): *Otherwise if $\text{Rank}(\cal{B})=k-\nu$, then any $(k-\nu)-$columns of $\cal{B}$ are linearly independent. From(2.39) it follows that,

[TABLE]

Construct $\widetilde{\cal{B}}$ by choosing any $(k-\nu)-$columns of $\cal{B}.$ Further, choose the corresponding $(k-\nu)$ eigenvalues of $\cal{D}_{2}$ and denote it by $\widetilde{\cal{D}_{2}}.$ Then

[TABLE]

Now since any $(k-\nu)-$eigenvalues of $\cal{D}_{2}$ is equal to the eigenvalues of $\cal{D}_{1}$, we get that all the eigenvalues of $\cal{D}_{1}$ and $\cal{D}_{2}$ should be equal. This proves that $\cal{D}_{1}=\rho\text{Id}_{k-\nu}$ and $\cal{D}_{2}=\rho\text{Id}_{\nu}.$ Note that since $\cal{A}$ and $\cal{D}$ are not invertible, $\cal{C}$ should have non-zero elements, i.e., $\delta^{\prime}_{-}=\delta_{+}=\eta.$ Further, $\cal{A}$ and $\cal{D}$ cannot be identically zero matrices, since that would mean $\text{Rank}(A)<k.$ Hence $\eta=\rho.$ This proves that for $|\eta|=1$,

[TABLE]

The similar argument will work if $\nu<k-\nu$ by interchanging the role of rows and columns of $\cal{B}.$

*Case 2: * For $\nu=k-\nu$, i.e., $\text{lcm}(\nu,k-\nu)=\nu$ and $\cal{S}=S^{\nu}$, $\cal{T}=T^{\nu}.$

So by chain rule

[TABLE]

Also

[TABLE]

Now

[TABLE]

Thus

[TABLE]

As before from (2.34) and (2.39), by analyzing all possible cases we again have

[TABLE]

where $|\delta_{\pm}|=1.$ This completes the proof of Theorem 1.1.

3. Dynamics of skew products of Hénon maps

In this section, we describe dynamics of the maps of the form (1.2). We consider the following three cases:

Case 1: $\lvert c\rvert>1$

Fix a $\delta\in\left(0,1\right)$. For $R>1$, recall that

[TABLE]

Lemma 3.1**.**

For any $\delta\in\left(0,h\right)$ where $h=\min\{\lvert c_{H}\rvert,1\}$, there exists $R_{0}=R_{0}(\delta)>1$ such that

[TABLE]

for all $R\geq R_{0}$. Further,

[TABLE]

for all $(\lambda,x,y)\in V_{R}^{+}$.

Proof.

We prove (3.1) by induction. Note that

[TABLE]

where $c_{jk}$’s are polynomials in $\mathbb{C}$ with degree at most $\tilde{d}$. Therefore, there exists $M>1$ such that

[TABLE]

for all $j,k$’s with $0\leq(j+k)\leq(d-1)$ where ${\lvert\lambda\rvert}_{+}=\max\{\lvert\lambda\rvert,1\}$. For $(\lambda,x,y)\in V_{R}^{+}$, combining (3.2) and (3.3), we get that

[TABLE]

The last inequality follows since $\lvert y\rvert>{\lvert\lambda\rvert}_{+}^{\tilde{d}+1}$ which in turn gives ${\lvert\lambda\rvert}_{+}^{\tilde{d}}<{\lvert y\rvert}^{\frac{\tilde{d}}{\tilde{d}+1}}$. Thus it follows from (3.4) that for any given $\delta\in\left(0,h\right)$ with $h=\min\{\lvert c_{H}\rvert,1\}$, one can choose $R$ sufficiently large such that

[TABLE]

for all $(\lambda,x,y)\in V_{R}^{+}$, which in turn gives

[TABLE]

in $V_{R}^{+}$. Now (3.5) gives that

[TABLE]

if $\left\lvert y\right\rvert>R$ with $R$ sufficiently large. Further, a similar calculation as in (3.4) gives that

[TABLE]

for $\lvert y\rvert$ sufficiently large. From (3.5) and (3.7), it follows that

[TABLE]

for $(\lambda,x,y)\in V_{R}^{+}$ with $R$ sufficiently large. Further, using (3.5), we get that

[TABLE]

for $(\lambda,x,y)\in V_{R}^{+}$. Thus if we take $\lvert y\rvert$ sufficiently large, then we get

[TABLE]

Combining (3.6), (3.8) and (3.9), we get that

[TABLE]

for $R$ sufficiently large.

Let (3.1) holds for $1\leq n\leq k$, i.e.,

[TABLE]

for $1\leq n\leq k$. Using (3.10), it follows that

[TABLE]

for $R>0$ sufficiently large and for all $n\geq 1$. Therefore, using (3.5), we get

[TABLE]

which, using induction hypothesis, gives

[TABLE]

This finishes the proof. ∎

Lemma 3.2**.**

For $(\lambda,x,y)\in K^{+}$, we have

[TABLE]

Proof.

Let $(\lambda,x,y)\in K^{+}$ which implies that $H^{n}(\lambda,x,y)=(c^{n}\lambda,x_{n}^{\lambda},y_{n}^{\lambda})\notin V_{R}^{+}$ for all $n\geq 0$, i.e.,

[TABLE]

Now

[TABLE]

for some $L,M>1$. The last inequality follows from (3.12). Further, note that

[TABLE]

Also if

[TABLE]

then it follows that

[TABLE]

which leads to a contradiction if $R$ is sufficiently large. Therefore

[TABLE]

and by induction we get

[TABLE]

∎

Lemma 3.3**.**

There exists a constant $K>1$ depending on the coefficients of $H$ such that for all $n\geq 1$ and for all $(\lambda,x,y)\in\mathbb{C}^{3}$, we have

[TABLE]

where {\lvert x\rvert}_{+}\big{/}{\lVert(x,y)\rVert}_{+}=\max\{{\lvert x\rvert}\big{/}{\lVert(x,y)\rVert},1\}, for $x,y\in\mathbb{C}$.

Proof.

Using (1.4), it follows that

[TABLE]

for some $K>1$. Further, inductively we get that

[TABLE]

where

[TABLE]

This finishes the proof. ∎

Theorem 3.4**.**

The sequence of functions $G_{n,H}^{\pm}$ converges to $G_{H}^{\pm}$ uniformly on compact sets in $\mathbb{C}^{3}$. The function $G_{H}^{\pm}$ is plurisubharmonic in $\mathbb{C}^{3}$ and pluriharmonic in $\mathbb{C}^{3}\setminus K_{H}^{\pm}$ satisfying

[TABLE]

in $\mathbb{C}^{3}$. Further, $G_{H}^{\pm}>0$ in $U_{H}^{\pm}$ and it vanishes precisely in $K_{H}^{\pm}$.

Proof.

It follows from Lemma 3.2 that $G_{n,H}^{+}\rightarrow 0$ as $n\rightarrow\infty$ in $K_{H}^{+}$. Further, it follows from Lemma 3.1 that

[TABLE]

in $U_{H}^{+}$. Note that (1.8) gives

[TABLE]

for some $C>1$, which in turn gives that $G_{n,H}^{+}$ converges uniformly to the function $G_{H}^{+}$ in $U_{H}^{+}$. Clearly, $G_{H}^{+}$ is pluriharmonic in $U_{H}^{+}$ and it follows from Lemma 3.1 that $G_{H}^{+}>0$ in $U_{H}^{+}$. Note that $U_{H}^{+}$ is completely invariant under $H$ which shows that the set $K_{H}^{+}$ is also completely invariant under $H$. It follows from (1.8) that

[TABLE]

in $\mathbb{C}^{3}$. Further, it follows from Lemma 3.1 that

[TABLE]

in $\mathbb{C}^{3}$ for each $n\geq 1$ where $M$ is some constant depending on $H$. Now let $G_{H*}^{+}$ is the upper semicontinuous regularization of $G_{H}^{+}$. So, $G_{H*}^{+}$ is plurisubharmonic in $\mathbb{C}^{3}$. Further, since $G_{H}^{+}$ is pluriharmonic in $U_{H}^{+}\cup{\rm{int}}(K_{H}^{+})$, it follows that $G_{H*}^{+}=G_{H}^{+}$ in $U_{H}^{+}\cup{\rm{int}}(K_{H}^{+})$. In addition, $G_{H*}^{+}$ enjoys the same functorial property as $G_{H}^{+}$ indicated in (3.13), i.e.,

[TABLE]

in $\mathbb{C}^{3}$ and using (3.14), we get

[TABLE]

for $(\lambda,x,y)\in K_{H}^{+}$. Now combining Lemma 3.3 and (3.15), we get that

[TABLE]

as $n\rightarrow\infty$ for all $(\lambda,x,y)\in K_{H}^{+}$. Therefore, $G_{H*}^{+}(\lambda,x,y)=0$ for $(\lambda,x,y)\in K_{H}^{+}$. Thus $G_{H*}^{+}=G_{H}^{+}$ in $\mathbb{C}^{3}$ which proves that $G_{H}^{+}$ is plurisubharmonic in $\mathbb{C}^{3}$. Now using Hartog’s Lemma we conclude that $G_{n,H}^{+}$ converges uniformly to $G_{H}^{+}$ on compact sets in $\mathbb{C}^{3}$. Running the similar set of arguments, we can prove the analogous result for $G_{H}^{-}$. ∎

Recall that

[TABLE]

Running the similar set of arguments as in Lemma 3.1, we get the following.

Lemma 3.5**.**

For any $\delta\in\left(0,h\right)$ where $h=\min\{\lvert c_{H}^{\prime}\rvert,1\}$, there exists $R_{0}=R_{0}(\delta)>1$ such that

[TABLE]

for all $R\geq R_{0}$. Further,

[TABLE]

for all $(\lambda,x,y)\in V_{R}^{-}$.

Lemma 3.6**.**

For each $(\lambda,x,y)\in K_{H}^{-}$, $G_{n,H}^{-}(\lambda,x,y)\rightarrow 0$ as $n\rightarrow\infty$.

Proof.

Let $(\lambda,x,y)\notin U_{H}^{-}$ which implies $H^{-n}(\lambda,x,y)=(c^{-n}\lambda,\tilde{x}_{n}^{\lambda},\tilde{y}_{n}^{\lambda})\notin V_{R}^{-}$ for any $n\in\mathbb{N}$. For a given $\lambda\in\mathbb{C}$, choose $n_{0}$ large enough such that $\lvert c^{-n_{0}}\lambda\rvert<1$. Now let

[TABLE]

for $n\geq n_{0}$. Now using (1.5), it follows that

[TABLE]

Note that $B_{jk}$’s are polynomials in $\mathbb{C}$ with degree at most $\tilde{d}$. Therefore, there exists $L>1$ such that

[TABLE]

for all $j,k$’s with $0\leq(j+k)\leq(d-1)$ where ${\lvert\lambda\rvert}_{+}=\max\{\lvert\lambda\rvert,1\}$. Now using (3.17), (3.18) and (3.19), we get that

[TABLE]

for some $L>1$ and for all $n\geq n_{0}$. Thus by induction, we get

[TABLE]

where $\hat{d}={d}/{d_{1}}$. Using the inequality (3.20), we get that $G_{n,H}^{-}(\lambda,x,y)\rightarrow 0$ as $n\rightarrow\infty$. ∎

Since $G_{H}^{\pm}$ are pluriharmonic in $\mathbb{C}^{3}\setminus K_{H}^{\pm}$ and in $\mathbb{C}^{3}\setminus J_{H}^{\pm}$, the following proposition holds. Note that the similar result holds for a single Hénon map (Proposition 3.8, [5]). Since, one can prove the following proposition using the same arguments as in the case of a single Hénon map, we are omitting the proof.

Proposition 3.7**.**

The functions $G_{H}^{\pm}$ are the pluricomplex Green functions of the sets $K_{H}^{\pm}$ and of the sets $J_{H}^{\pm}$.

Proposition 3.8**.**

For $(\lambda,x,y)\in V_{R}^{+}$,

[TABLE]

as $n\rightarrow\infty$ and for $(\lambda,x,y)\in V_{R}^{-}$,

[TABLE]

as $n\rightarrow\infty$.

Proof.

Since $H(V_{R}^{+})\subseteq V_{R}^{+}$, using (1.4) we get that

[TABLE]

for some $L>1$. Also from (3.5), it follows that for $\delta>0$ small enough,

[TABLE]

Combining (3.21) and (3.22), it follows that

[TABLE]

for some $K>1$. Now since $\lvert y\rvert>R>1$, it follows from (3.23) that

[TABLE]

as $n\rightarrow\infty$. Using the similar arguments, one can show that for $(\lambda,x,y)\in V_{R}^{-}$

[TABLE]

as $n\rightarrow\infty$. ∎

Remark 3.9*.*

Therefore it follows from Proposition 3.8 that if $(\lambda,x,y)\in V_{R}^{+}$, then $H^{n}(\lambda,x,y)\rightarrow[0:0:1:0]$ in $\mathbb{P}^{3}$ as $n\rightarrow\infty$. Similarly, if $(\lambda,x,y)\in V_{R}^{-}$, then $H^{-n}(\lambda,x,y)\rightarrow[0:1:0:0]$ in $\mathbb{P}^{3}$ as $n\rightarrow\infty$.

Case 2: $\lvert c\rvert<1$ and Case 3: $\lvert c\rvert=1$

The analogues of the above results can be proved for the skew products of Hénon maps with $\lvert c\rvert\leq 1$ and to do so one needs to consider the cases $\lvert c\rvert<1$ and $\lvert c\rvert=1$ separately. In particular, Remark 3.9 holds for all $0\neq c\in\mathbb{C}$. Recall that we need to modify the filtrations in each cases.

4. Rigidity of skew products of Hénon maps with non-comapct parameter space

Böttcher coordinates and its relation with Green functions

Proposition 4.1**.**

Let $H:\mathbb{C}^{3}\rightarrow\mathbb{C}^{3}$ be a skew products of Hénon maps. Then there exist non-vanishing holomorphic functions $\phi_{H}^{\pm}:V_{R}^{\pm}\rightarrow\mathbb{C}$ such that

[TABLE]

in $V_{R}^{+}$ and

[TABLE]

in $V_{R}^{-}$. Further,

[TABLE]

and

[TABLE]

Proof.

Consider the following telescoping product

[TABLE]

Now by (1.4),

[TABLE]

Hence,

[TABLE]

Now

[TABLE]

for $\lvert y\rvert>R>>1$ and for all $n\geq 1$. The second last inequality follows from (1.4) and from the fact that $H(V_{R}^{+})\subseteq V_{R}^{+}$. The last inequality follows from (3.1).

Let ${\left(1+\frac{q_{c^{n}\lambda}(x_{n}^{\lambda},y_{n}^{\lambda})}{c_{H}{(y_{n}^{\lambda})}^{d}}\right)}^{\frac{1}{\tilde{d}d^{n}}}$ be the principal branch of $\tilde{d}d^{n}$-th root of ${\left(1+\frac{q_{c^{n}\lambda}(x_{n}^{\lambda},y_{n}^{\lambda})}{c_{H}{(y_{n}^{\lambda})}^{d}}\right)}$. Now note that the convergence of the product in (4.1) is equivalent to the convergence of the series

[TABLE]

Since, (4.2) holds, the above series in (4.3) is absolutely convergent which shows that the product in (4.1) is convergent.

Since the infinite product in (4.1) is convergent, we can define the function $\phi_{H}^{+}:V_{R}^{+}\rightarrow\mathbb{C}$ as follows:

[TABLE]

Since

[TABLE]

which we get combining (3.1) and (3.4), it follows that

[TABLE]

as $\lVert(\lambda,x,y)\rVert\rightarrow\infty$ in $V_{R}^{+}$.

Note that

[TABLE]

Hence,

[TABLE]

where $L(\lambda,x,y)\rightarrow 0$ as $\lVert(\lambda,x,y)\rVert\rightarrow\infty$ in $V_{R}^{+}$. Thus

[TABLE]

Now $H^{-n}(\lambda,x,y)=(c^{-n}\lambda,\tilde{x}_{n}^{\lambda},\tilde{y}_{n}^{\lambda})$ where $\tilde{x}_{n}^{\lambda}$ is a polynomial in $x$ and $y$ of degree $\tilde{d}d^{n-1}$ which is precisely the degree of $H^{-n}$. Consider the following telescoping product

[TABLE]

which can be shown to be convergent as before. We define

[TABLE]

That

[TABLE]

as $\lVert(\lambda,x,y)\rVert\rightarrow\infty$ in $V_{R}^{-}$ and

[TABLE]

in $V_{R}^{-}$, can be shown in the similar fashion as in the case of $\phi_{H}^{+}$. ∎

Further, it follows from (1.8) that

[TABLE]

in $V_{R}^{+}$ and

[TABLE]

in $V_{R}^{-}$.

Proof of the Theorem 1.2:

Since $J_{H}^{\pm}=J_{F}^{\pm}$, it follows from Proposition 3.7 that the pluricomplex Green functions of these sets coincide, i.e.,

[TABLE]

At this point, we refer the readers to the proof of Theorem 1.1 in [8] (or Theorem 1.1 in the present paper) which can be adapted to show that

[TABLE]

in $\mathbb{C}^{3}$ for some $\delta_{+},\ \delta_{-}\in\mathbb{C}$ with $\lvert\delta^{\pm}\rvert=1$. Note that Proposition 5.6 along with the relations (4.5) and (4.6) play a crucial role to establish (4.7). This finishes the proof.

5. Rigidity of skew products of Hénon maps with compact parameter space

In this section, we consider skew products of Hénon maps fibered over a compact metric space $M$, i.e., the maps of the form

[TABLE]

for $(\lambda,x,y)\in M\times\mathbb{C}^{2}$ where $\sigma_{H}$ acts as an homeomorphism on $M$. Recently, the dynamics of these maps has been studied in [13] and [14]. For each $\lambda\in M$ and for each $n\geq 1$, let

[TABLE]

and

[TABLE]

We define the non-escaping sets and escaping sets as follow:

[TABLE]

and

[TABLE]

respectively. Further, for each $\lambda\in M$, we define $J^{\pm}_{H,\lambda}=\partial K^{\pm}_{H,\lambda}$. Further, define

[TABLE]

for each $n\geq 1$ and for all $(x,y)\in\mathbb{C}^{2}$. Recall form [14] that the sequence of functions defined in (5.2) converge uniformly in compact to the plurisubharmonic functions $G_{H,\lambda}^{\pm}$ in $\mathbb{C}^{2}$. Further, note that (see [14]) a uniform filtration defined as follow:

[TABLE]

with $R>0$ sufficiently large works uniformly for all $H_{\lambda}$’s.

5.1. Fibered Böttcher coordinates

For each $\lambda\in M$, let

[TABLE]

be a composition of generalized Hénon maps as described before. Clearly, the degree of ${(H_{\lambda})}_{1}$ is strictly less than that of ${(H_{\lambda})}_{2}$ when considered as polynomials in $y$. Further, let, for each $\lambda\in M$

[TABLE]

where $c_{H}=\Pi_{j=1}^{m}c_{j}^{d_{(j+1)}\cdots d_{m}}$ with $d_{j+1}\cdots d_{m}=1$ when $j=m$, $d=d_{n}\cdots d_{1}$ and $q_{\lambda}$ a polynomial in $x,y$ of degree strictly less than $d$.

Similarly, let

[TABLE]

where degree of $(H_{\lambda}^{2})^{\prime}$ is strictly less than that of $(H_{\lambda}^{1})^{\prime}$ considering as polynomials in $x$. As before, one can write

[TABLE]

where $c_{H}^{\prime}=\Pi_{j=1}^{m}(c_{j}\delta_{j}^{-1})^{d_{(j-1)}\cdots d_{1}}$ where $d_{j-1}\cdots d_{1}=1$ when $j=1$ and $q_{\lambda}^{\prime}$ a polynomial in $x,y$ of degree strictly less than $d$.

Proposition 5.1**.**

Let $H$ be a skew products of Hénon maps of the form (5.1), then for each $\lambda\in M$, there exist non-vanishing analytic functions $\phi_{H,\lambda}^{\pm}:V_{R}^{\pm}\rightarrow\mathbb{C}$ such that

[TABLE]

in $V_{R}^{+}$ and

[TABLE]

in $V_{R}^{-}$. Further,

[TABLE]

and

[TABLE]

Proof.

Let,

[TABLE]

for $(x,y)\in\mathbb{C}^{2}$ and

[TABLE]

are polynomials in $x$ and $y$ in $\mathbb{C}^{2}$. Further, note that the degree of $y_{n,\lambda}$ is $d^{n}$.

Now consider the following telescoping series

[TABLE]

Note that form (5.3), we get the followings:

[TABLE]

Since the parameter space $M$ is compact and and the coefficients of $q_{\lambda}$’s vary continuously in $\lambda$, we can choose an $R>0$ sufficiently large such that

[TABLE]

for all $\lvert y\rvert>R$ and for all $n\geq 1$. Hence, ${\left(1+\frac{q_{\sigma_{H}^{n}(\lambda)}(x_{n,\lambda},y_{n,\lambda})}{c_{H}y_{n,\lambda}^{d}}\right)}^{\frac{1}{d^{n+1}}}$, the principal branch of the $d^{n+1}$-th root of ${\left(1+\frac{q_{\sigma_{H}^{n}(\lambda)}(x_{n,\lambda},y_{n,\lambda})}{c_{H}y_{n,\lambda}^{d}}\right)}$ is well-defined for all $\lambda\in M$ and for all $n\geq 1$.

Note that, for a fixed $\lambda\in M$, there exist $L,\ \tilde{L}>0$ such that

[TABLE]

for $R>0$ large enough. Now since, convergence of the series

[TABLE]

implies the convergence of the series

[TABLE]

using (5.7), we conclude that the series (5.9) converges and consequently the series (5.8) also converges.

Therefore,

[TABLE]

exists.

Now for each $\lambda\in M$, define $\phi_{H,\lambda}^{+}:V_{R}^{+}\rightarrow\mathbb{C}$ as follows:

[TABLE]

Since

[TABLE]

it follows that

[TABLE]

where $L_{\lambda}(x,y)\rightarrow 0$ as $\lVert(x,y)\rVert\rightarrow\infty$ in $V_{R}^{+}$.

Now

[TABLE]

Since $L_{\lambda}(x,y)\rightarrow 0$ as $\lvert y\rvert\rightarrow\infty$ in $V_{R}^{+}$ and $c_{H}^{\frac{1}{d^{n}}}\rightarrow 1$ as $n\rightarrow\infty$, we get

[TABLE]

in $V_{R}^{+}$.

As before, we define

[TABLE]

a polynomial in $x$ and $y$ of degree $d^{n}$ with leading term ${c}_{H}^{\prime}$. Next, define

[TABLE]

which makes sense for the same reason explained in the previous case.

Using a same sort of arguments as in (5.11), we get that

[TABLE]

for all $(x,y)\in V_{R}^{-}$ and for all $\lambda\in M$. ∎

5.2. Rigidity of skew products of Hénon maps with the same fibered Julia sets

Proposition 5.2**.**

Let $H$ and $F$ be skew products of Hénon maps fibered over a compact metric space $M$ and let for each $\lambda\in M$, the fibered Julia sets of $H$ and $F$ are the same, i.e., $J_{H,\lambda}^{\pm}=J_{F,\lambda}^{\pm}$ for all $\lambda\in M$. Then for each $\lambda\in M$,

[TABLE]

for some $\gamma:(x,y)\mapsto(\delta^{+}x,\delta^{-}y)$ with $\lvert\delta^{\pm}\rvert=1$. Consequently, there exists a skew map $\Gamma:(\lambda,x,y)=(\mu(\lambda),\delta^{+}x,\delta^{-}y)$ such that $F\circ H=\Gamma\circ H\circ F$ in $M\times\mathbb{C}^{2}$, where

[TABLE]

and $\lvert\delta^{\pm}\rvert=1$.

Proof.

Let

[TABLE]

and

[TABLE]

Since for each $\lambda\in M$, the functions $G_{H,\lambda}^{+}$ and $G_{F,\lambda}^{+}$ are the pluricomplex Green functions for the sets $J_{H,\lambda}^{+}$ and $J_{F,\lambda}^{+}$ respectively (see Proposition 1.2 in [13]), we have

[TABLE]

for all $\lambda\in M$. Thus combining (5.10) and (5.2), we get that

[TABLE]

and

[TABLE]

in $V_{R}^{+}$ for all $\lambda\in M$. The rest of the proof follows by using fibered Böttcher coordinates constructed in the Proposition 5.1 and pursuing the same techniques as in the proof of Theorem 1.1 in [8].

∎