Rigidity Theorems for H\'{e}non maps-II

TL;DR

This paper investigates the rigidity properties of Hénon maps, showing that sharing certain dynamical invariants implies commutativity of their squares and identical non-escaping sets, and establishes local injectivity of key associations.

Contribution

It proves that equal Green measures, filled Julia sets, or Green functions imply the commutativity of squared Hénon maps and the equality of non-escaping sets, extending understanding of their rigidity.

Findings

Hénon maps with the same Green measure have commuting squares.

Hénon maps with the same filled Julia set have commuting squares.

Associations of Hénon maps to their Green measure, Julia set, or Green function are locally injective.

Abstract

The purpose of this note is to explore further the rigidity properties of H\'{e}non maps from arXiv:1806.08189. For instance, we show that if $H$ and $F$ are H\'{e}non maps with the same Green measure ($\mu_H=\mu_F$), or the same filled Julia set ($K_H=K_F$), or the same Green function ($G_H=G_F$), then $H^2$ and $F^2$ have to commute. This, in turn, gives that $H$ and $F$ have the same non-escaping sets. Further we prove that, either of the association of a H\'{e}non map $H$ to its Green measure $\mu_H$ or to its filled Julia set $K_H$ or to its Green function $G_H$ is locally injective.

Equations

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

$$H=H_m\circ H_{m-1}\circ \cdots \circ H_1$$

$$H_j(x,y)=(b_{j}y+c_j,p_j(y)-{\delta }_{j}x)$$

$$K_H^{\pm }=\{z\in {\mathbb{C}}^2:H^{\pm n}(z)\text{ is bounded for every }n\ge 1\}.$$

$$F\circ H=C\circ H\circ F\text{ or }{F}^{-1}\circ H=C\circ H\circ F^{-1}$$

$$V_R^{+}$$

$$V_R^{-}$$

$$V_R$$

$$H(V_R^{+})\subset V_R^{+},H(V_R^{+}\cup V_R)\subset V_R^{+}\cup V_R$$

$$H^{-1}(V_R^{-})\subset V_R^{-},H^{-1}(V_R^{-}\cup V_R)\subset V_R^{-}\cup V_R.$$

$${\mathbb{C}}^2\setminus K_H^{\pm }=\bigcup _{n=0}^{\infty }(H^{\mp n})(V_R^{\pm })$$

$$G_H^{\pm }(x,y)=\underset{n\to \infty }{\lim }\frac{1}{d^n}{\log }^{+}\Vert H^{\pm n}(x,y)\Vert .$$

$$G_H^{\pm }\circ H(z)=d{G}_H^{\pm }(z)$$

$$G_H^{+}(x,y)=\log |y|+O(1)$$

$$G_H^{-}(x,y)=\log |x|+O(1).$$

$$J_H^{\pm }=\partial K_H^{\pm }.$$

$$J_H=J_H^{+}\cap J_H^{-}\text{ and }{K}_H=K_H^{+}\cap K_H^{-}.$$

$${\mu }_H^{\pm }=\frac{1}{2\pi }d{d}^c{G}_H^{\pm }$$

$$\mu_{H}=\mu^{+}_{H}\wedge\mu^{-}_{H}=\Big{(}\frac{1}{2\pi}dd^{c}G_{H}\Big{)}^{2}.$$

$$F^2\circ H^2=H^2\circ F^2.$$

$$H^2=H_m\circ \cdots \circ H_1$$

$$F\circ H=C_{\eta }\circ H\circ F=H\circ F\circ C_{\eta }\text{ or }F\circ H=C_{\eta }\circ H\circ F=H\circ F\circ C_{\eta }^{-1}.$$

$${\varphi }_H^{+}\circ H(x,y)=c_H{({\varphi }_H(x,y))}^{d_H}\text{ and }{\varphi }_F^{+}\circ F(x,y)=c_F{({\varphi }_F(x,y))}^{d_F}$$

$${\varphi }_H^{-}\circ H^{-1}(x,y)=c_H^{\prime }{({\varphi }_H^{-}(x,y))}^{d_H}\text{ and }{\varphi }_F^{-}\circ F^{-1}(x,y)=c_F^{\prime }{({\varphi }_F(x,y))}^{d_F}$$

$${\varphi }_F^{+}(x,y),{\varphi }_H^{+}(x,y)\sim y\text{ as }\Vert (x,y)\Vert \to \infty \text{ in }{V}_R^{+}$$

$${\varphi }_F^{-}(x,y),{\varphi }_H^{-}(x,y)\sim x\text{ as }\Vert (x,y)\Vert \to \infty \text{ in }{V}_R^{-}.$$

$$G_H^{+}=\log |{\varphi }_H^{+}|+\frac{1}{d_H-1}\log |c_H|=\log |{\varphi }_F^{+}|+\frac{1}{d_F-1}\log |c_F|=G_F^{+}$$

$$G_H^{-}=\log |{\varphi }_H^{-}|+\frac{1}{d_H-1}\log \left|c_H^{\prime }\right|=\log |{\varphi }_F^{-}|+\frac{1}{d_F-1}\log |c_F^{\prime }|=G_F^{-}$$

$$\frac{1}{d_H-1}\log |c_H|=\frac{1}{d_F-1}\log |c_F|$$

$${\varphi }_H^{+}\equiv {\varphi }_F^{+}$$

$${\varphi }_H^{-}\equiv {\varphi }_F^{-}$$

Full text

1. Introduction

We continue to explore the rigidity properties of Hénon maps from [5]. The main motivation for [5] was a result from the dynamics of polynomial maps in one variable by Beardon [1], namely if two polynomials $P$ and $Q$ of degree greater than or equal to $2$, have the same Julia set, i.e., $J_{P}=J_{Q}$ then

[TABLE]

where $\sigma(z)=az+b$ with $|a|=1$ and $\sigma(J_{P})=J_{P}.$ In [5], we provide an analogue of this result for Hénon maps in $\mathbb{C}^{2}.$ To explain it in detail, let us revisit the notations first. Let $\cal{H}$ denote the collection of maps $H$ defined as:

[TABLE]

where each $H_{j}$ is a map of the form

[TABLE]

with $p_{j}$ a polynomial of degree $d_{j}\geq 2$, $b_{j}\delta_{j}\neq 0$ and $c_{j}\in\mathbb{C}.$ The degree of $H$ is $d=d_{1}d_{2}\ldots d_{m}$. The phrase Hénon map, hereafter will be used to refer a map as in (1.1). The definition of Hénon map deviates from the standard definition of Hénon map in normal form, as we allow the constants $b_{j}\in\mathbb{C}^{*}$ and $c_{j}\in\mathbb{C}.$ The reason, we work with this definition is the presence of this constants do not make any difference from the point of view of dynamics.

Recall that for a Hénon map $H$, the non–escaping sets or the filled positive and negative Julia sets of $H$ is defined as:

[TABLE]

Theorem 1.1 of [5] says that, if an automorphism $F$ preserves the non–escaping sets of a Hénon map, i.e., $F(K_{H}^{\pm})=K_{H}^{\pm}$ then $F$ or $F^{-1}$ is a Hénon map of the form (1.1) and further, they commute upto a linear map. In particular,

[TABLE]

where $C(x,y)=(\delta_{+}x,\delta_{-}y)$ with $|\delta_{\pm}|=1.$ Though in [5] we did not explicit mention about the presence of the constants $c_{j}$’s in $H$, but there presence do not significantly change any computation (see Lemma 3.1). The proof of this result crucially relied on techniques developed by Buzzard and Fornæss in [6], the rigidity theorem of Dinh and Sibony in [7], and introducing the notion of Böttcher coordinates for Hénon maps of the form (1.1), inspired from the construction of Hubbard and Oberste-Vorth in [10].

The goal of this paper is to improve the rigidity phenomenon for Hénon maps further. We briefly recall the dynamical objects associated to a Hénon map $H\in\cal{H}$ from [2] and [3]. For $R>0$, let

[TABLE]

For a given $H\in\mathcal{H}$ there exists $R>0$ such that

[TABLE]

and

[TABLE]

Also, $K^{\pm}\subset V_{R}\cup V^{\mp}_{R}$ and

[TABLE]

The positive and negative Green functions associated to the Hénon map $H$ is defined as:

[TABLE]

The Green functions $G_{H}^{\pm}$ is pluri–subharmonic on $\mathbb{C}^{2}$ and non–negative everywhere, pluri–harmonic on $\mathbb{C}^{2}\setminus K^{\pm}_{H}$ and vanish precisely on $K^{\pm}_{H}$. By construction the Green functions satisfy

[TABLE]

and they have a logarithmic growth near infinity, i.e., there exists $R>0$, sufficiently large such that for $(x,y)\in V_{R}\cup V_{R}^{+}$

[TABLE]

and for $(x,y)\in V_{R}\cup V_{R}^{-}$

[TABLE]

The Green function of $H$ is defined as $G_{H}=\max\{G_{H}^{+},G_{H}^{-}\}.$ Then $G_{H}$ is a pluri–subharmonic function with logarithmic growth in $\mathbb{C}^{2}$. The forward and the backward Julia sets associated to $H$ is defined as:

[TABLE]

Further we define the sets $J_{H}$ and $K_{H}$ as follows:

[TABLE]

Henceforth, by Julia set we will mean the set $J_{H}$ and by filled Julia set we will mean the set $K_{H}.$ It turns out that both $J_{H}$ and $K_{H}$ are compact sets, which are completely invariant under $H.$ Also, $G_{H}^{\pm}$ are the pluri–complex Green functions for $K^{\pm}_{H}$ respectively and $G_{H}$ the pluri–complex Green function of $K_{H}$. The supports of the positive closed $(1,1)$ currents

[TABLE]

are $J^{\pm}_{H}.$ The Green measure of $H$ is defined as:

[TABLE]

Also, $\mu_{H}$ is the equillibrium measure of $K_{H}$ which is invariant under $H$ and $H^{-1}.$ Thus for two Hénon maps $H$ and $F$, if $K_{H}=K_{F}$ then $\mu_{H}=\mu_{F}.$

In Section 2, we consider the simplest form of a Hénon map, i.e., $H(x,y)=(y,p(y)-ax)$ and prove the following rigidity result:

Theorem 1.1**.**

Suppose $H(x,y)=(y,p(y))-ax)$ and $F(x,y)=(y,q(y)-bx)$ be two (simple) Hénon maps. Then each of the following are true.

• (i)

If $F(K_{H}^{\pm})=K_{H}^{\pm},$ then $F=H.$

• (ii)

If $\mu_{H}=\mu_{F},$ then $F=H.$

• (iii)

If $K_{H}=K_{F},$ then $F=H.$

• (iv)

If $G_{H}=G_{F},$ then $F=H.$

Note that Theorem 1.1 is an analogue of Beardon’s result ([1]) for simple Hénon maps. To prove this, we use facts about the uniqueness of pluri–subharmonic functions with logarithmic growth from [4] and conclude that any two Hénon maps with the same Green measure have the same Green function. Next, we explore the dynamical properties of a Hénon map in appropriate regions and prove that the positive and negative Green functions of these Hénon maps actually coincide on $V_{R}^{\pm}$, respectively for appropriately chosen large $R.$ Finally, by appealing to techniques similar to the proof of Theorem 1.1 in [5] the result follows.

In Section 3, we first establish a rigidity result similar to Theorem 1.1 from [5], which is stated as follows:

Theorem 1.2**.**

Let $F$ be a non–linear automorphism that preserves $K_{H}^{\pm}$ where $H$ is a Hénon map then $F$ or $F^{-1}$ is a Hénon map and

[TABLE]

To prove this, without loss of generality first we assume that origin is fixed by the Hénon map $H$, and prove that $H^{2}$ can be uniquely expressed (upto composition by linear maps) in the normal form such that

[TABLE]

where $H_{i}(x,y)=(y,p_{i}(y)-\delta_{i}x)$ and $H_{i}(0)=0$ for every $1\leq i\leq m.$ Note that as a consequence of Theorem 1.1 from [5], $F$ and $H$ satisfy (1.3). Now by further analyzing closely the behavior of the linear map $C$, it is possible to remove the linear map in the second iterate of $H$ and $F.$ In this context, we also provide an explicit example to emphasize that $H$ and $F$ might fail to commute even though their squares commute. Further appealing to ideas from Section 2, we prove a version of Theorem 1.1, particularly for Hénon maps, which is stated as follows:

Theorem 1.3**.**

Let $F$ and $H$ are Hénon maps such that either $K_{H}=K_{F}$ or $\mu_{H}=\mu_{F}$ or $G_{H}=G_{F}.$ Then $K_{H}^{\pm}=K_{F}^{\pm}$ and $G_{H}^{\pm}=G_{F}^{\pm}.$

Observe that Theorem 1.2 applied to Theorem 1.3, gives that $H^{2}$ and $F^{2}$ actually commutes. To mention here, Lamy in [11] proved that if two Hénon maps $H$ and $F$ have the same positive Green function, i.e., $G_{H}^{+}=G_{F}^{+}$ then there exist integers $m$ and $n$ such that $F^{m}=H^{n}.$ Note that Theorem 1.3 improves this result, in the sense if $H$ and $F$ are Hénon maps such that $G_{H}=G_{F}$, (or $K_{H}=K_{F}$, or $\mu_{H}=\mu_{F}$) then there exist positive integers $m$ and $n$ such that $F^{m}=H^{n}.$ Also, Dujardin and Favre in connection to their work on Manin–Mumford problem for plane polynomial automorphisms, in [8], used number theoretic techniques to show that, if $F$ and $G$ are polynomial automorphisms of Hénon type of the affine plane over a number field that share a Zariski dense set of periodic points, then there exist positive integers $m$ and $n$ such that $F^{m}=G^{n}.$

Finally, we consider the space of Hénon maps $\cal{H}$ with the topology of uniform convergence over compact sets and consider either of the associations – $H$ to its Green measure or $H$ to its Green function or $H$ to its filled Julia set, i.e., $H\to G_{H}$ or $H\to\mu_{H}$ or $H\to K_{H}$ on $\cal{H}.$ By using Theorem 1.2 and the fact that, roots of a Hénon map are finite from [6] we prove that this association is locally injective in either of the cases. The statement of the theorem is stated as:

Theorem 1.4**.**

Let $\mathcal{H}$ denote the space Hénon maps then the mappings $H\to G_{H}$, $H\to\mu_{H}$ and $H\to K_{H}$ is locally injective on $\mathcal{H}.$

Acknowledgements

The author would like to thank Ratna Pal and Kaushal Verma for introducing her to the problem.

2. Proof of Theorem 1.1

In this section, we first prove a development to Theorem 1.1 from [5].

Theorem 2.1**.**

Let $H$and $F$ be Hénon maps of the form (1.1) such that $F$ preserves $K_{H}^{\pm}$ ($F(K_{H}^{\pm})=K_{H}^{\pm}$) then there exists an $\eta$ such that $|\eta|=1$ and

[TABLE]

where $C_{\eta}(x,y)=(\eta x,\eta^{-1}y).$

Proof.

Let $d_{H}$ denote the degree of $H.$ From Proposition 2.1 of [5], there exists $R>0$ (sufficiently large), appropriate non–zero constants $c_{H},c^{\prime}_{H},c_{F},c^{\prime}_{F}\in\mathbb{C}^{*}$ and non–vanishing holomorphic functions (Böttcher coordinates) $\phi_{H}^{\pm},\phi_{F}^{\pm}:V_{R}^{\pm}\rightarrow\mathbb{C}$ such that

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

Also, from the proof of Theorem 1.1 in [5] the positive and negative Green function of $F$ and $H$ coincides, i.e., $G_{H}^{\pm}=G_{F}^{\pm}$ and

[TABLE]

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

[TABLE]

in $V_{R}^{-}.$ Since $\phi_{H}^{+}$ and $\phi_{F}^{+}$ are both asymptotic to $y$ as $\lVert(x,y)\rVert\rightarrow\infty$ in $V_{R}^{+}$, it follows that

[TABLE]

and consequently

[TABLE]

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

[TABLE]

in $V_{R}^{-}$. From now on we shall write $\phi^{\pm}$ for $\phi_{H}^{\pm}\equiv\phi_{F}^{\pm}$.

By (2.5),

[TABLE]

for some $\delta_{i}$ ($i=1,2$) with $\lvert\delta_{i}\rvert=1$.

*Step 1: * There exist $\delta_{1},\tilde{\delta}_{1}\in\mathbb{C}^{*}$ such that in $\mathbb{C}^{2}$

[TABLE]

where $|\delta_{1}|$ and $\tilde{\delta}_{1}|=1.$

By (2.1),

[TABLE]

and similarly,

[TABLE]

Therefore,

[TABLE]

on $V_{R}^{+}$. Since

[TABLE]

and

[TABLE]

as $\lVert(x,y)\rVert\rightarrow\infty$ in $V_{R}^{+}$, it follows that for a fixed $x_{0}\in\mathbb{C}$,

[TABLE]

The expression on the left is a polynomial in $y$ and hence

[TABLE]

for all $y\in\mathbb{C}$. Therefore,

[TABLE]

Now note that $F\circ H(V_{R}^{+})\cap H\circ F(V_{R}^{+})\neq\phi$, since $[0:1:0]$ is an attracting fixed point of both $F$ and $H$ in $\mathbb{P}^{k}.$ Let $\cal{U}=F\circ H(V_{R}^{+})\cap H\circ F(V_{R}^{+})$ (an open subset $V_{R}^{+}$). From (2.7) and (2.8) it follows that for $(x,y)\in\cal{U}$

[TABLE]

[TABLE]

and similarly,

[TABLE]

Hence

[TABLE]

on $\cal{U}$ with $|\eta|=1$. Consequently, there exists $\tilde{\delta}_{1}$ (an appropriate $d_{H}d_{F}-$th root of $\eta$) such that

[TABLE]

on $\cal U$. Note that $|\tilde{\delta}_{1}|=1$. Pick a point $c\in\mathbb{C}^{*}$ and a sequence $\{y_{n}\}\in\mathbb{C}$ such that $|y_{n}|\to\infty.$ The points

[TABLE]

as $n\to\infty,$ i.e., $(c,y_{n})\in\cal{U}$ for $n$ sufficiently large. Note that $(F^{-1}\circ H^{-1})(c,y_{n})$, and $(H^{-1}\circ F^{-1})(c,y_{n})$, is contained in $V_{R}^{+}$ as a consequence of $(c,y_{n})$ being contained in $\mathcal{U}$. Hence we have,

[TABLE]

as $n\rightarrow\infty$.

Since $\phi^{+}(x,y)\sim y$ as $\lVert(x,y)\rVert\rightarrow\infty$,

[TABLE]

as $n\rightarrow\infty$. The expression on the left is a polynomial in $y$ for each fixed $c$ and thus

[TABLE]

for all $y\in\mathbb{C}$. Using the same argument as in the previous case, we get

[TABLE]

Thus Step 1 is complete from (2.9) and (2.10).

*Step 2: * There exist $\delta_{2},\tilde{\delta}_{2}\in\mathbb{C}^{*}$ such that in $\mathbb{C}^{2}$

[TABLE]

where $|\delta_{2}|$ and $\tilde{\delta}_{2}|=1.$

Using similar idea as in Step 1, by interchanging the role of $F$ with $F^{-1}$ and $H$ with $H^{-1}$ and working with the function $\phi^{-1}$ in $V_{R}^{-}$, Step 2 follows.

Thus from (2.6) and (2.11) we have that

[TABLE]

where $C_{1}(x,y)=(\delta_{2}x,\delta_{1}y)$ and $C_{2}(x,y)=(\tilde{\delta}_{2}x,\tilde{\delta}_{1}y).$ Now (2.12) gives

[TABLE]

*Step 3: * $C_{1}(x,y)=C_{2}(x,y)$ or $C_{1}(x,y)=C_{2}^{-1}(x,y)$ and $C_{1}(x,y)=C(x,y)=(\eta x,\eta^{-1}y).$

Let $A=DH(F(0))DF(0).$ Then by chain rule applied to (2.13)

[TABLE]

where $D_{2}=\text{Diag}(\tilde{\delta}_{2},\tilde{\delta}_{1})$ and $D_{2}=\text{Diag}({\delta}_{2},{\delta}_{1}).$ Since $A$ is invertible, one of the diagonal element and one of the off–diagonal element cannot be simultaneously zero. Hence this leads to two generic situation:

• (i)

Either both the diagonal elements should be non–zero. This gives $\tilde{\delta}_{1}=\delta_{1}$ and $\tilde{\delta}_{2}=\delta_{2}$(from (2.14)).

• (ii)

Or both off–diagonal elements should be non–zero. This gives $\delta_{2}=\tilde{\delta}_{1}$ and $\tilde{\delta}_{2}=\delta_{1}$ (from (2.14)).

Further, as Hénon maps have a constant Jacobian it follows (2.13) that

[TABLE]

Hence

[TABLE]

Thus the proof. ∎

Let $H$ be a Hénon map of the form(1.1), i.e.,

[TABLE]

where $H_{i}$ are Hénon maps of the form (1.2). Recall that, there exists $R_{H}>0$ for which $V^{\pm}$ can be defined as:

[TABLE]

such that $H_{i}(V^{+})\subset V^{+}$, $H_{i}^{-1}(V^{-})\subset V^{-}$ for every $1\leq i\leq n.$ For $R_{0}>R_{H}$ consider the region $D^{1}(R_{0})\subset V^{-}$ and $D^{2}(R_{0})\subset V^{+}$ defined as:

[TABLE]

and

[TABLE]

Proposition 2.2**.**

Let $H$ be a Hénon map of the form (1.1), i.e.,

[TABLE]

where $H_{i}$’s are of the form (1.1) for every $1\leq i\leq m.$ Then there exist $R_{H}>0$ and $R_{0}\gg R_{H}$ such that

• (i)

on $D^{2}(R_{0})$, $G_{H}^{-}<G_{H}^{+}.$

• (ii)

on $D^{1}(R_{0})$, $G_{H}^{+}<G_{H}^{-}.$

Proof.

Let $d_{i}\geq 2$ be the degree of each $H_{i}$, $1\leq i\leq m$. By assumption $V^{\pm}=V^{\pm}_{R_{H}}$ and $H_{i}(V^{+})\subset V^{+}$, $H^{-1}_{i}(V^{-})\subset V^{-}$ for every $1\leq i\leq m.$ Further there exist constants $m$ and $M$, such that on $V^{+}$

[TABLE]

and on $V^{-}$

[TABLE]

Patching up (2.15) and (2.16) for every $i$, there exist constants $C_{1}$ and $C_{2}$ such that

[TABLE]

where $d=d_{m}\ldots d_{1}.$ Suppose $H_{1}(x,y)=(b_{1}y+c_{1},p_{1}(y)-\delta_{1}x)$ then

[TABLE]

Let $C=\max\{|b_{1}^{-1}(x-c_{1})|:|x|<R_{H}\}.$ Choose $R_{1}>\max\{C,R_{H}\}$ such that for $|y|>R_{1}$,

[TABLE]

whenever $|x|<R_{H}$. Clearly, this means that

[TABLE]

Thus from the assumptions on $R_{H}$ and (2.18) for every $n\geq 1$,

[TABLE]

Let

[TABLE]

Now from (2.15) and (2.16) there exist constants $C_{3}$ and $C_{4}$ such that

[TABLE]

for $(x,y)\in D^{2}(R_{1}).$ So at the $n-$th iterate from (2) we have the following:

[TABLE]

From (2.19) and (2) it follows that

[TABLE]

Hence from (2) the constants $C_{1}$, $C_{2}$, $C_{3}$ and $C_{4}$ can be appropriately modified to obtain the following:

[TABLE]

Note that on $V^{+}$,

[TABLE]

and on $V^{-}$,

[TABLE]

Dividing (2) with $d^{n}$ and taking limit $n\to\infty$ it follows that

[TABLE]

and

[TABLE]

Since $(x,y)\in D^{2}(R_{1})$, there exists $K>1$ such that

[TABLE]

Thus from (2.23) and (2.24) there exists a real constant $M$ such that

[TABLE]

for $|y|$ sufficiently large and $(x,y)\in D^{2}(R_{1}).$ Further modifying the choice of $R_{1}$ to $R_{0}^{+}$ (sufficiently large) it follows that $G_{H}^{-}<G_{H}^{+}$ on $D^{2}(R_{0}^{+}).$

Similarly there exists $R_{0}^{-}$ such that $G_{H}^{+}<G_{H}^{-}$ on $D^{1}(R_{0}^{-}).$ Now choose $R_{0}=\max\{R_{0}^{+},R_{0}^{-}\}$ and this completes the proof. ∎

Theorem 2.3**.**

Let $H$ and $F$ be two Hénon maps such that the Green current associated to $H$ and $F$ are same, i.e., $\mu_{H}=\mu_{F}.$ Then

[TABLE]

Proof.

Let $G_{H}$ and $G_{F}$ denote the Green function associated to $F$ and $H.$ Then

[TABLE]

By Theorem 1 in [4], we have that $G_{H}=G_{F}.$

Let

[TABLE]

where $H_{j}$ and $F_{i}$ are Hénon maps of the form (1.2). There exists $R>0$ such that if $V^{\pm}=V_{R}^{\pm}$ then

[TABLE]

for every $1\leq i\leq m$ and $1\leq j\leq n.$

By Proposition 2.2 there exist $R_{H}^{-}$, $R_{F}^{+}$ and $R_{F}^{-}$ such that $G_{H}^{+}<G_{H}^{-}$ on $D^{1}(R_{H}^{-})$, $G_{F}^{+}<G_{F}^{-}$ on $D^{2}(R_{F}^{+})$ and $G_{F}^{+}<G_{F}^{-}$ on $D^{1}(R_{F}^{-}).$ Let $R_{0}=\max\{R_{H}^{+},R_{H}^{-},R_{F}^{+},R_{F}^{-}\}.$ Since

[TABLE]

it follows that on $D^{2}(R_{0})$

[TABLE]

and on $D^{1}(R_{0})$

[TABLE]

Also, $G_{H}^{+}$ and $G_{H}^{-}$ are harmonic on $V_{R}^{+}$ and $V_{R}^{-}$ respectively, hence $G_{H}^{+}=G_{F}^{+}\text{ on }V_{R}^{+}$, and $G_{H}^{-}=G_{F}^{-}\text{ on }V_{R}^{-}.$

Now note that in the proof of Theorem 2.1, we only use the fact that the positive Green functions of $F$ and $H$ agree on $V^{+}$ and negative Green functions of $F$ and $H$ agree in $V^{-}.$ Hence by using exactly the same arguments as in the proof of Theorem 2.1 it follows that there exists $\eta$ such that

[TABLE]

where $C_{\eta}(x,y)=(\eta x,\eta^{-1}y).$ ∎

Remark 2.4*.*

The fact that, Green measure is the equillibrium measure supported on the filled Julia set of a Hénon map together with the proof of Theorem 2.3 implies that for two Hénon maps $H$ and $F$, $\mu_{H}=\mu_{F}$ is equivalent to $G_{H}=G_{F}$ is equivalent to $K_{H}=K_{F}.$

Proof of Theorem 1.1.

By Theorem 2.1, Theorem 2.3 and Remark 2.4 in either of these cases there exists a $C_{\eta}$ such that

[TABLE]

Note that in this case the first diagonal element of $A=DF(H(0))DH(0)$ is always non–zero, forcing $\delta_{1}=\tilde{\delta}_{1}.$ Thus

[TABLE]

From $F\circ H=C_{\eta}\circ H\circ F$, by equating the first coordinate it follows that

[TABLE]

Also, $F^{-1}\circ H^{-1}=C_{\eta}\circ H^{-1}\circ F^{-1}$. By comparing the second coordinate we have

[TABLE]

This is not possible for $\eta\neq 1$, hence the proof. ∎

3. Proof of Theorems 1.2, 1.3 and 1.4

In this section, we will further improve Theorem 2.1. We say that a Hénon map $H$ can be expressed in normal form if

[TABLE]

where each $H_{i}(x,y)=(y,p_{i}(y)-\delta_{i}x)$ such that degree of $p_{i}\geq 2$ and $\delta_{i}\neq 0.$

Lemma 3.1**.**

Given a Hénon map of the form (1.1), say $H$,

[TABLE]

can be expressed in normal form.

Proof.

By assumption, $H=H_{m}\circ\cdots\circ H_{1}$ where $H_{i}$’s are Hénon map of the form (1.2), i.e.,

[TABLE]

such that $\delta_{i}b_{i}\neq 0$ and $d_{i}$, the degree of $p_{i}$ is greater than or equal to $2.$ Then

[TABLE]

and $H^{2}$ is an even composition of Hénon maps of the form (1.2). Now

[TABLE]

Define $\tilde{H}_{i}(x,y)=(y,\tilde{p}_{i}(y)-\tilde{\delta}_{i}x)$ and $\tilde{H}_{i+1}(x,y)=(y,\tilde{p}_{i+1}(y)-\tilde{\delta}_{i+1}x)$ where

[TABLE]

and

[TABLE]

Note that

[TABLE]

and hence the proof. ∎

Lemma 3.2**.**

Let $H$ be a Hénon map such that

[TABLE]

where $m\geq 2$ and $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x)$ for every $1\leq i\leq m.$ Then $H$ can be expressed in normal form as a $m-$composition.

Proof.

We will prove by induction on $m.$ For $m=2$, i.e., $H=H_{2}\circ H_{1}$ the proof is same as Lemma 3.1, by taking $c_{1}=c_{2}=0.$ Assume that the statement is true for some $m\geq 2$, and prove for $m+1$, i.e.,

[TABLE]

consider $\tilde{H}_{m+1}(x,y)=(y,p_{m+1}(y)-\delta_{m+1}x)$ and $\tilde{H}_{m}(x,y)=(b_{m}y,b_{m+1}p_{m}(y)-b_{m+1}\delta_{m}x)$ then

[TABLE]

By induction hypothesis, $\tilde{H}_{m+1}^{-1}\circ H$ can be expressed in normal form and thus the proof. ∎

Lemma 3.3**.**

Let $H$ be a Hénon map of the form (1.1) which can be expressed in the normal form as:

[TABLE]

where $H_{i}(x,y)=(y,p_{i}(y)-\delta_{i}x)$ and $p_{i}(0)=0$ for every $1\leq i\leq m.$ Further if there exists $C_{\eta}$, such that

[TABLE]

for some $|\eta|=1.$ Then $\eta p_{i}(\eta y)=p_{i}(y)$ for every $1\leq i\leq m.$

Proof.

The above statement is always true for $\eta=1.$ So assume $\eta\neq 1$. Then

[TABLE]

Let $A_{m}=H_{m}^{-1}C_{\eta}H_{m}.$ Note that $A_{m}$ should be a linear map of the form $(ax+c,dx+ey+f)$, otherwise there will be an inconsistency in the highest degree of $y$ on both the sides of (3.2). Since

[TABLE]

by computing $A_{m}$ we have that

[TABLE]

Thus $\delta_{m}^{-1}(p_{m}(\eta y)-\eta^{-1}p_{m}(y))$ should be a constant (say $C_{m}$). Since $p_{m}(0)=0$, it follows that $C_{m}=0$, i.e., $\eta p_{m}(\eta y)=p_{m}(y).$ Also, (3.2) reduces to

[TABLE]

Now by inductive argument the proof follows. ∎

Remark 3.4*.*

Note that from the proof of Lemma 3.3, it follows that, for every $1\leq i\leq m$

[TABLE]

Lemma 3.5**.**

Let $H$ be a Hénon map such that $H(0)=0.$ Then $H$ can be represented as:

[TABLE]

where $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x)$ and $p_{i}(0)=0$ for every $1\leq i\leq m.$

Proof.

Recall from Corollary 2.3 in [9] the expression of a Hénon map is unique upto composition by linear elementary maps. In particular, if

[TABLE]

where $H_{i}$ and $\tilde{H}_{i}$ are Hénon maps of the form (1.2). Then $m=\tilde{m}$ and $H_{i}=s_{i}\circ\tilde{H}_{i}$ or $H_{i}=\tilde{H}_{i}\circ s_{i}$ where $s_{i}(x,y)=(ax+b,cy+d).$ Hence, we can use induction on the number of Hénon maps (here it is $m$) of the form (1.2) composed to obtain the given Hénon map.

*Initial case: *For $m=1$, since $H(0)=0$, it follows that $H_{1}=\tilde{H}_{1}$ and the statement is true.

*Induction Statement: *Suppose the statement is true for $m-1$, i.e., if $H$ is a Hénon map such that $H(0)=0$ obtained by $(m-1)-$composition in normal form. Then $H$ can be expressed as:

[TABLE]

where $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x)$ and $p_{i}(0)=0$ for every $1\leq i\leq m-1.$

*General case: * Thus

[TABLE]

where $\tilde{H}_{i}(x,y)=(b_{i}y+c_{i},\tilde{p}_{i}(y)-\delta_{i}x)$ for every $1\leq i\leq m.$ Define, $p_{1}(x)=\tilde{p}_{1}(x)-\tilde{p}(0)$ and let

[TABLE]

Now

[TABLE]

Now redefine $\tilde{H}_{2}=\tilde{H}_{2}\circ A.$ Then $\tilde{H}$ is a Hénon map such that $\tilde{H}(0)=0$ obtained by

$(m-1)-$composition in normal form. By Induction hypothesis, $\tilde{H}$ can be expressed as:

[TABLE]

where $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x)$ and $p_{i}(0)=0$ for every $2\leq i\leq m.$ Thus the proof. ∎

Remark 3.6*.*

Suppose $H$ is a Hénon map such that $H(0)=0$ and

[TABLE]

where $m\geq 2$ and $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x).$ From Lemma 3.2 and Lemma 3.5, $H$ can be expressed in the normal form, i.e.,

[TABLE]

where $\tilde{H}_{i}(x,y)=(y,\tilde{p}_{i}(y)-\tilde{\delta}_{i}x)$ for $1\leq i\leq m.$ Note that for every $1\leq i\leq m$, there exist non-zero constants $m_{i}$ and $M_{i}$ such that

[TABLE]

Hence, for some $|\eta|=1$, if $\eta\tilde{p}_{i}(\eta y)=$ then $\eta p_{i}(\eta y)=p_{i}(y).$

Now we can complete the proof of Theorem 1.2.

Proof of Theorem 1.2.

From Theorem 1.1 of the [5], we know that $F$ should be a polynomial map, in particular either $F$ or $F^{-1}$ is a Hénon map of the form (1.1).

By Theorem 2.1, it follows that there exist $\eta_{1}$ and $\eta_{2}$ with $|\eta_{i}|=1$ for $i=1,2$ such that

[TABLE]

and

[TABLE]

where $C_{\eta_{i}}(x,y)=(\eta_{i}x,\eta^{-1}_{i}y).$

*Case 1: * If $H(0)=0.$

*Subcase 1: *If $F(0)\neq 0$ then from (3.3)

[TABLE]

Thus proving $C_{\eta_{1}}=\text{Identity}$ and $H^{2}\circ F^{2}=F^{2}\circ H^{2}.$

*Subcase 2: * If $F(0)=0$ then by Lemma 3.5, the expression of $F$ and $H$ can be modified such that

[TABLE]

where $F_{j}(x,y)=(\beta_{j}y,q_{j}(y)-\rho_{j}x)$ and $H_{i}(x,y)=(b_{i}y,p_{i}(y)-\delta_{i}x)$ with $p_{i}(0)=0$ and $q_{j}(0)=0$ and $b_{i}\delta_{i},\beta_{j}\rho_{j}\neq 0$ for every $1\leq i\leq m$ and $1\leq j\leq n.$

From (3.3) there exists $C_{\eta_{1}}$ such that

[TABLE]

From Lemma 3.2, $H\circ F$ can be expressed in the normal form. Further from Remark 3.4 and 3.6, if any $p_{i}$’s or $q_{j}$’s have a linear term then

[TABLE]

If $F\circ H=H\circ F$ then clearly $F^{2}\circ H^{2}=H^{2}\circ F^{2}.$ Otherwise, if $F\circ H=-H\circ F$, then

[TABLE]

Now

[TABLE]

Suppose none of the $p_{i}$’s or $q_{j}$’s have a linear term then

[TABLE]

Thus $DF^{2}(0)$ and $DH^{2}(0)$ are diagonal matrices. Since $H^{2}(0)=0$ and $F^{2}(0)=0$,

[TABLE]

But from (3.4)

[TABLE]

thus proving $\eta_{2}=1.$ Hence $F^{2}\circ H^{2}=H^{2}\circ F^{2}.$

*Case 2: * If $H(0)\neq 0$, then consider any fixed point of the Hénon map, say $p=(p_{1},p_{2}).$ Let $A_{p}$ be the affine map translating [math] to $p$, i.e.,

[TABLE]

Define $\tilde{H}=A_{p}^{-1}\circ H\circ A_{p}$ and $\tilde{F}=A_{p}^{-1}\circ F\circ A_{p}.$ The both $\tilde{H}$ and $\tilde{F}$ are Hénon maps and $\tilde{H}(0)=0.$

Claim: $K_{\tilde{H}}^{\pm}=A_{p}^{-1}(K_{H}^{\pm})$ and $\tilde{F}(K_{\tilde{H}}^{\pm})=K_{\tilde{H}}^{\pm}.$

For $z\in A_{p}^{-1}(K_{H}^{+})$, $\tilde{H}^{n}(z)$ is bounded for every $n$, thus $A_{p}^{-1}(K_{H}^{+})\subset K_{\tilde{H}}^{+}.$ Similarly, for $z\in A_{p}(K_{\tilde{H}}^{+})$, $H^{n}(z)$ is bounded for every $n$ proving $A_{p}(K_{\tilde{H}}^{+})\subset K_{H}^{+}.$ By an exactly similar argument for $H^{-1}$ and $K_{H}^{-}$ we have

[TABLE]

Further as $F(K_{H}^{\pm})=K_{H}^{\pm}$,

[TABLE]

Now by Case 1,

[TABLE]

Finally, if $F^{-1}$ is a Hénon map then by the above arguments we have

[TABLE]

∎

In the following example we show that Theorem 1.2 is optimal, in the sense there exist Hénon maps $H$ and $F$ such that $F(K_{H}^{\pm})=K_{H}^{\pm}$ but they do no commute.

Example 3.7*.*

Let $H(x,y)=(y,y^{2}-x)$ and $F(x,y)=(\omega y,(\omega y)^{2}-\omega^{2}x)$ where $\omega$ is the cube root of unity. Note that

[TABLE]

Hence $F$ is a Hénon map such that $F(K_{H}^{\pm})=K_{H}^{\pm}.$ Now

[TABLE]

thus $H$ and $F$ do not commute.

Finally, we prove Theorem 1.3 and 1.4 using Theorem 1.2.

Proof of Theorem 1.3.

Recall that, by Remark 2.4, it is enough to prove for the case $\mu_{H}=\mu_{F}.$ Further from Theorem 2.3, it follows that

[TABLE]

where $C_{\eta}(x,y)=(\eta x,\eta^{-1}y).$ Now by applying exactly similar argument as in the proof of Theorem 1.2, it follows that

[TABLE]

For $z\in K_{H}^{+}$

[TABLE]

Thus $F^{2}(z)\in K_{H}^{\pm}$, or $F^{2}(K_{H}^{+})\subset K_{H}^{+}.$ Similarly $F^{-2}(K_{H}^{+})\subset K_{H}^{+}$. Also, applying the same arguments with $H^{-2n}$ it follows that $F^{2}(K_{H}^{-})=K_{H}^{-}.$ Hence

[TABLE]

Now for $z\in K_{H}^{+}$, $F^{2n}(z)\in V_{R}\cup V_{R}^{-}$ and $z\in K_{F}^{+}$, $H^{2n}(z)\in V_{R}\cup V_{R}^{-}$ proving

[TABLE]

By a similar argument for $H^{-1}$ and $F^{-1}$ it follows that $K_{H}^{-}=K_{F}^{-}$. Hence $K_{F}^{\pm}=K_{H}^{\pm}$ and $G_{H}^{\pm}=G_{F}^{\pm}.$ ∎

Proof of Theorem 1.4.

Again, by Remark 2.4 it is sufficient to prove the statement only for the Green measure of $H$. Suppose $\{H_{n}\}$ is any sequence of Hénon maps such that $\mu_{H_{n}}=\mu_{H}$ and $H_{n}\neq H.$ To prove the result, it is enough to prove that no subsequence of $\{H_{n}\}$ converges to $H$ in the topology of uniform convergence over compact subsets of $\mathbb{C}^{2}$. In other words, there exist a neighbourhood $U_{H}$ around $H$ in $\cal{H}$ with the aforementioned topology, such that the map $H\to\mu_{H}$ is injective on $U_{H}.$

We will prove the result by contradiction. Suppose not, i.e., there exists a subsequence $\{H_{n_{k}}\}$ of $\{H_{n}\}$ such that $H_{n_{k}}\to H$ as $k\to\infty.$ With abuse of notation, we denote $\{H_{n_{k}}\}$ by $\{H_{n}\}.$

Claim: $A_{n}=H_{n}^{-1}\circ H$ should be linear maps for $n-$sufficiently large.

Let $d_{n}$ be the degree of $H_{n}$ and $d$ the degree of $H.$ By Theorem 1.3, $G_{H_{n}}^{\pm}=G_{H}^{\pm}$ and $K_{H_{n}}^{\pm}=K_{H}^{\pm}.$ Fix a $z\notin K_{H}^{+}$,

[TABLE]

By assumption $A_{n}(z)\to z$ as $n\to\infty$ and $G_{H}^{+}(A_{n}(z))\to G_{H}^{+}(z).$ Thus $d_{n}d^{-1}\to 1$ as $n\to\infty$, i.e., $d_{n}=d$ for $n-$sufficiently large. Now if $A_{n}$’s are non–linear then either $d_{n}<d$ or $d<d_{n}$, which is not true. Hence $A_{n}$’s are linear and by Theorem 1.1 from [5]

[TABLE]

By Theorem 1.2, $H_{n}^{2}\circ H^{2}=H^{2}\circ H_{n}^{2}$, hence

[TABLE]

where $C_{n}=H_{n}^{2}\circ H^{-2}$ and $k\geq 1.$ Note that by an argument similar to $A_{n}$’s, $C_{n}$’s are also linear maps of the form 3.5. Let

[TABLE]

*Case 1: * Suppose $H(0)=0.$ Note that the number of fixed point of $H^{2}$ should be finite. Let $S_{H^{2}}$ denote the set of fixed points of $H^{2}$, i.e.,

[TABLE]

From (3.6) it follows that $C_{n}(0)\in S_{H^{2}}$, hence $(b_{n},d_{n})\in S_{H^{2}}.$ Thus there are only finitely many choice for $b_{n}$’s and $d_{n}$’s. Now by Lemma 3.1 and Lemma 3.5

[TABLE]

such that $m\geq 2$ and $\tilde{H}_{i}(x,y)=(y,p_{i}(y)-\delta_{i}x)$ where $p_{i}(0)=0$ and degree of $p_{i}$, say $d_{i}\geq 2$ for every $1\leq i\leq m.$ Note that

[TABLE]

Hence $\tilde{H}_{m}^{-1}\circ C_{n}\circ\tilde{H}_{m}$ should be a linear map of the (3.5), otherwise there will be an inconsistency in the degree of $y$ in both the sides. This means

[TABLE]

for some $d^{\prime}_{n}\in\mathbb{C}.$ Thus from (3.7) and (3.8) it follows that

[TABLE]

By a similar argument for $\tilde{H}_{m-1}\circ\tilde{C}_{n}\circ\tilde{H}_{m-1}$ it follows that $c_{n}^{d_{m-1}}=a_{n}.$ Thus both $a_{n}$’s and $c_{n}$’s should be $(d_{m}d_{m-1}-1)-$th root of unity and there can be only a finitely many choice for both $a_{n}$’s and $c_{n}$’s. So there are only finitely many possible choice for the elements in the sequence $\{C_{n}\}$ and hence for $H_{n}^{2}$’s. Now as

[TABLE]

it follows that $H_{n}^{2}=H^{2}$ for $n$ sufficiently large (say $n_{0}$). Thus $H_{n}$’s should be a square root of $H^{2}$ for all $n\geq n_{0}.$ By Theorem 4.1 in [6], there can only be a finitely many choice for $H_{n}$’s again. Hence, $H_{n}=H$ for $n\geq\tilde{n}_{0}$ (sufficiently large). This completes the proof for Case 1.

*Case 2: *Suppose $H(0)\neq 0$. Let $p_{0}$ be a fixed point of $H$, consider $\tilde{H}=A_{p_{0}}^{-1}\circ H\circ A_{p_{0}}$ where $A_{p_{0}}(z)=z+p_{0}$, for $z\in\mathbb{C}^{2}.$ By assumption $H_{n}\to H$, hence

[TABLE]

as $n\to\infty.$ Now by Case 1, $A_{p_{0}}^{-1}\circ H_{n}\circ A_{p_{0}}=\tilde{H}$, i.e., $H_{n}=H$ for $n$ sufficiently large. This completes the proof. ∎