Dynamics of semigroups of Hénon maps

In this article, we study the dynamics of a semigroup of polynomial automorphisms of $\mathbb{C}^{2}$, generated by finitely many Hénon maps. To explain the setup, let $\mathscr{G}$ be a given finite collection of automorphisms of $\mathbb{C}^{2}$. We will consider the semigroup $\mathcal{S}$ generated by the elements of $\mathscr{G}$ under the composition operation, which will be denoted as

and $n_{0}\geq 1$, a positive integer. Furthermore, we assume the maps $\mathsf{H}_{i}$ are Hénon maps, i.e., for every $1\leq i\leq n_{0}$, there exists $m_{i}\geq 1$ such that

and $H_{j}^{i}(x,y)$ is a map of the form

where $a\neq 0$ and $p$ is a polynomial of degree at least 2, for every $1\leq i\leq n_{0}$ and $1\leq j\leq m_{i}$.

Recall from [26], a map of the above form (1.3) was termed as generalised Hénon map, and as Hénon if $p(y)=y^{2}+c$, classically. However, the maps of form (1.2) are more general than (1.3) and the methods to study maps of form (1.3), mostly generlises for the entire class. Hence by the terminology Hénon map we will mean maps of the form (1.2). Our interest to study the dynamics of semigroups generated by Hénon maps, is for the following facts

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  Firstly, a classical result of Friedland–Milnor [17] states that these maps are essentially the only class of polynomial automorphisms of $\mathbb{C}^{2}$, exhibiting interesting (iterative) dynamics and have been studied intensively. For instance, see [21], [4], [12] etc. Further, their dynamics are known to be connected to the dynamics of polynomial maps in $\mathbb{C}$ (see [6]) and they also extend as (bi)rational maps of $\mathbb{P}^{2}.$

• •

  Secondly, the study of dynamics of arbitrary families, i.e., non-iterative families of holomorphic self-maps (endomorphisms) is important from the point of view of complex function theory. In particular, the (non-autonomous) basins of attraction — see Theorem 1.1 below for definition — of a sequence of automorphisms of $\mathbb{C}^{k}$, $k\geq 2,$ with a common attracting fixed point has lead to the construction of pathological domains in $\mathbb{C}^{k}$ (see [28], [1], [15]).

• •

  Also, it is conjectured — follows as a consequence of a conjecture, originally due to Bedford ([3],[16]) — that a non-autonomous basin of attraction of sequences of automorphisms of $\mathbb{C}^{2}$, that vary within an infinite (or finite) collection sharing a common uniformly attracting fixed point should be biholomorphic to $\mathbb{C}^{2}$. The same is true for autonomous (or iterative) basins of endomorphisms with an attracting fixed point (see [28, Theorem 9.1]).

To mention here in Section 7, we answer the above for a finite collection of Hénon maps by the methods developed in this article to study the semigroup $\mathcal{S}$. It is stated as

Further, we study a few particular cases of an infinite collection or parametrised families of Hénon maps sharing a common attracting fixed point with ‘uniform bounds’. In particular, the following example is obtained as a consequence of Corollary 7.4 (also see Example 7.5), in comparison to Theorem 1.4 and 1.10 in [15].

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  Lastly, the study of dynamics of rational semigroups on $\mathbb{P}^{1}$ is an interesting and widely studied area. This setup was introduced by Hinkkanen–Martin, in [20], motivated by their connection to the dynamics of Kleinian groups on the Riemann sphere, observed in [18].

Our primary goal in this article is, to explore the dynamics of a semigroup of Hénon maps both in $\mathbb{P}^{2}$ and $\mathbb{C}^{2}$, motivated by the study of dynamics of rational semigroups in $\mathbb{P}^{1}.$ In particular, we will attempt to connect between results from iterative dynamics of Hénon maps of the form (1.2) and semigroup dynamics of rational maps in $\mathbb{P}^{1}$ to the current setup. Later, we will generalise a few results appropriately in the setup of non-autonomous families to obtain the aforementioned applications.

Let $X$ be a complex manifold and $\mathcal{S}$ be an infinite family of holomorphic self-maps of $X.$ The Fatou set for the family $\mathcal{S}$ is the largest open set of $X$ where the family $\mathcal{S}$ is normal, i.e.,

The Julia set $\mathcal{J}_{\mathcal{S}}$, is the complement of the Fatou set in $X.$

As reported earlier, the setup considering $X=\mathbb{P}^{1}$ and $\mathcal{S}$, a semigroup generated by more than one rational map of degree at least 2, was introduced in [20] and later on has been explored extensively. A major difficulty in this framework — as compared to the iterative dynamics — is neither the Julia set nor the Fatou set is completely invariant, in general.

It is a classical result of Brolin [8] that says - if $\mathcal{S}$ is the semigroup of iterates of a (single) polynomial map $p$ of degree at least 2, the limiting distribution of points in the preimages of a generic point $z\in\mathbb{P}^{1}$, corresponds to the equilibrium measure of the Julia set. Further, the potential associated with this measure, i.e., the Green’s function of the Julia set can be constructed via the dynamics of $p$. The equidistribution of the iterated preimages of a generic point $z\in\mathbb{P}^{1}$, i.e., the limiting distribution is independent of the (generic) $z$, was established for the iterations of rational map of $\mathbb{P}^{1}$ by Lyubich in [25]. Boyd in [7], extended Lyubich’s method and constructed an equidistributed measure supported on the Julia set of a finitely generated semigroup of rational maps (of degree at least 2) in $\mathbb{P}^{1}$. For a finitely generated semigroup of polynomials of degree at least 2, Boyd’s measure is not, in general, the equilibrium measure of its Julia set. Recently, in [24] the latter measure is interpreted as an equilibrium measure in the presence of an external field, which is given by a generalisation of the Greens function — attributed as ‘dynamical Greens function’.

To note, equidistributed measures exist for dynamics of certain meromorphic correspondences on compact connected Kähler manifolds, of appropriate intermediate degree (see [10]). However, birational maps of $\mathbb{P}^{2}$ obtained from extension of Hénon maps do not belong to the above category. Also, for iterative families of a Hénon map the Julia set is captured via the support of a unique positive closed $(1,1)$-current of mass 1, obtained by the action of $dd^{c}$-operator on the pluri-complex Green’s function of the Julia set. Furthermore, it is an equidistributed current in $\mathbb{C}^{2}$, in the sense, that it can be recovered as a limit of appropriately weighted preimages of an algebraic variety in $\mathbb{C}^{2}$ — see [4, Theorem 4.7] or [12, Corollary 6.7].

To mention here, construction of currents for non-autonomous families of Hénon maps have been done on an appropriate bounded region containing the origin, in [11], via the fact they are horizontal. Here, we construct a (similar) global equidistributed dynamical Green’s current on $\mathbb{P}^{2}$ in Corollaries 5.8 and 6.11 — using the dynamical Green’s functions, constructed by generalising ideas from [7], [11], [24] — both for the semigroups $\mathcal{S}$ and non-autonomous families. Thus, obtaining the uniqueness of the currents upto a semi-invariance property for the semigroups $\mathcal{S}$, stated in Corollary 1.6. Also, see Remark 6.12, for details.

Let us first recall a few important properties of iterations of a Hénon map $\mathsf{H}$. The pluri-complex Green’s function (see [22] for the definition) associated to the Julia set of iterates of $\mathsf{H}$ or $\mathsf{H}^{-1}$, say $G_{\mathsf{H}}^{+}$ or $G_{\mathsf{H}}^{-}$ respectively, can be recovered via the dynamics. In particular, if $d_{\mathsf{H}}$ is the degree of the map $\mathsf{H}$ then

where $\log^{+}x=\max\{\log x,0\}$ for every $x>0$ and $\|\cdot\|$ be the supremum norm in $\mathbb{C}^{2}$. Also, $\mu_{\mathsf{H}}^{\pm}$ are the closed positive $(1,1)$-(equidistributed) currents as mentioned previously. Note that the above definition holds for any norm on $\mathbb{C}^{2}$, however for the sake of convenience we will use the notation $\|\cdot\|$ to denote the supremum norm, throughout this article.

Now, let $\mathcal{S}$ be the semigroup as introduced in (1.1), i.e., $\mathcal{S}=\langle\mathscr{G}\rangle$ where $\mathscr{G}=\{\mathsf{H}_{i}:1\leq i\leq n_{0}\}$ and $\mathsf{H}_{i}$ are Hénon maps of the form (1.2) and of degree $d_{i}\geq 2$ for every $1\leq i\leq n.$ We first generalise a few definitions and observe some basic results regarding the semigroup $\mathcal{S}$ in Section 2. Particularly, we note that $\mathcal{S}$ might have more than one generating set, however it has a unique minimal generating set.

In Section 3, we generalise the construction of positive and negative Green’s functions, i.e., the functions $G_{\mathsf{H}}^{\pm}$ noted above, in the setup of the semigroup $\mathcal{S}.$ To do the same, we define the total degree of the semigroup $\mathcal{S}$ with respect to the generating set $\mathscr{G}$ as $D_{\mathscr{G}}=\sum_{i=1}^{n_{0}}d_{i}$, and consider the sequence of plurisubharmonic functions $G_{k}^{\pm}$ on $\mathbb{C}^{2}$ defined as

where $\mathscr{G}_{k}$ denote the elements of the semigroup $\mathcal{S}$ of length $k$ with respect to the generating set $\mathscr{G}$, i.e., $\mathscr{G}_{k}=\{\mathsf{H}_{i_{1}}\circ\cdots\circ\mathsf{H}_{i_{k}}:1\leq i_{j}\leq n_{0}\text{ and }1\leq j\leq k\}.$ We prove that the pointwise limits of the sequences $\{G_{k}^{\pm}\}$ constructed in (1.5) exist, which is stated as

Henceforth, the functions $G_{\mathscr{G}}^{\pm}$ will be referred as the dynamical positive (or negative) Green’s function associated to the semigroup $\mathcal{S}$ generated by the set $\mathscr{G}=\{{\mathsf{H}}_{i}:1\leq i\leq n_{0}\}$. The need to specify the generating set $\mathscr{G}$ is important as the semigroup $\mathcal{S}$ may admit multiple generating sets. Also, note that the functions $G_{\mathscr{G}}^{\pm}$ satisfy the following semi-invariance relation right by the construction (1.5) and Theorem 1.3.

Thus, as consequences of the proof of Theorem 1.3, we note that the functions $G_{\mathscr{G}}^{\pm}$ admit logarithmic growth on appropriate regions, and the strong filled positive and negative Julia sets of the semigroup $\mathcal{S}$ are pseudoconcave sets (see Corollary 3.4 and Remark 3.5).

Next, we analyse the Julia sets $\mathcal{J}_{\mathcal{S}}^{\pm}$ and the properties the dynamical Green’s $(1,1)$-currents associated to the functions $G_{\mathscr{G}}^{\pm}$, defined as $\mu_{\mathscr{G}}^{\pm}=\frac{1}{2\pi}dd^{c}G_{\mathscr{G}}^{\pm}$. Consequently, in Section 4, we prove the analogue to Corollary 2.1 from [20] — an important fact from the dynamics of semigroups of the rational maps on $\mathbb{P}^{1}$ — via the supports of $\mu_{\mathscr{G}}^{\pm}.$

Thus from Theorem 1.3 and the above, $G_{\mathscr{G}}^{\pm}$ is actually pluriharmonic on the Fatou sets $\mathcal{F}_{\mathcal{S}}^{\pm}$. In Section 5, we study the extension of the currents $\mu_{\mathscr{G}}^{\pm}$ to $\mathbb{P}^{2}$ and prove that they are limits of (weighted) equidistributed projective varieties, in the spirit of [12, Theorem 6.2]. Also, consequently we observe the following uniqueness of $\mu_{\mathscr{G}}^{+}$ upto a semi-invariance property.

In Section 6, we consider the dynamics of a non-autonomous sequence of Hénon maps, say $\{h_{k}\}\in\mathcal{S}$ and prove that there exist plurisubharmonic and continuous dynamical (positive and negative) Green’s functions, denoted by $\mathsf{G}_{\{h_{k}\}}^{\pm}$, with logarithmic growth. Thus $\mu_{\{h_{k}\}}^{\pm}=\frac{1}{2\pi}dd^{c}(\mathsf{G}_{\{h_{k}\}}^{\pm})$, are positive $(1,1)$-currents of mass 1, supported on the positive and negative Julia sets of the sequence $\{h_{k}\}.$ Also, we obtain the analogs to the equidistribution results, i.e., Corollaries 5.8 and 5.10, in this setup of non-autonomous dynamics. However, we will discuss them briefly as most of the ideas are similar to that realised in Section 5 and, depends upon the existence of the dynamical Green’s function with suitable growth at infinity.

Finally, we study the non-autonomous attracting basin of a sequence of Hénon maps of the form (1.2), admitting a uniformly attracting behaviour, on a neighbourhood of the origin. Further, we prove Theorem 1.1, as an application of the existence of Green’s functions $\mathsf{G}^{\pm}_{\{h_{k}\}}$ and enlist a few more applications, which follows from the technique. All of these affirmatively answers a few particular cases of the equivalent formulation of the Bedford Conjecture, in $\mathbb{C}^{2}$ for Hénon maps — as alluded to in the beginning.

In this section, we first observe a proposition about the generating set $\mathscr{G}$ of the semigroup $\mathcal{S}$ as in (1.1), which might not be unique, always. Recall the setup from Section 1, let $\displaystyle\mathscr{G}=\{\mathsf{H}_{i}:1\leq i\leq n_{0}\}$ where $\mathsf{H}_{i}$’s are Hénon maps of the form (1.2), with degree $d_{i}\geq 2$. The total degree of the semigroup $\mathcal{S}$ with respect to the generating set $\mathscr{G}$ is $D_{\mathscr{G}}=\sum_{i=1}^{n_{0}}d_{i}$ and $\mathscr{G}_{k}$ is the set of all elements of length $k$, $k\geq 1$ in the semigroup $\mathcal{S}$ with respect to $\mathscr{G}$.

Next, we revisit and introduce a few important definitions (and notations) with respect to the dynamics of the semigroup $\mathcal{S}$, that are independent of the generating set $\mathscr{G}$.

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  Let $\mathcal{S}^{-}$ denote the semigroup of maps comprising of the inverse of the maps that belong to $\mathcal{S}$ and $\mathscr{G}^{-}$ the inverse of the elements that belong to $\mathscr{G}$, i.e.,

  $$\mathscr{G}^{-}=\{\mathsf{H}_{i}^{-1}:1\leq i\leq n_{0}\}\text{ and }\mathcal{S}^{-}=\langle\mathscr{G}^{-}\rangle.$$

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  The Fatou sets of $\mathcal{S}$ and $\mathcal{S}^{-}$) — as stated in Section 1 — is denoted by $\mathcal{F}_{\mathcal{S}}^{+}$ and $\mathcal{F}_{\mathcal{S}}^{-}$ respectively. The positive and negative Julia sets are denoted by $\mathcal{J}_{\mathcal{S}}^{\pm}.$

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  We consider the following two alternatives for the filled positive and negative Julia sets.

  1. (1)

     The strong positive (or negative) filled Julia set is defined as the collection of all the points $z\in\mathbb{C}^{2}$ such that for every sequence $\{h_{k}\}\subset\mathcal{S}$, the sequence $\{h_{k}(z)\}$ (or the sequence $\{h_{k}^{-1}(z)\}$, respectively) is bounded, i.e.,

     $\displaystyle{\mathcal{K}}^{+}_{\mathcal{S}}=\big{\{}z\in\mathbb{C}^{2}:\text{for every sequence }\{h_{k}\}\subset\mathcal{S}\text{ the sequence }\{h_{k}(z)\}\text{ is bounded}\big{\}},$

     $\displaystyle{\mathcal{K}}^{-}_{\mathcal{S}}=\big{\{}z\in\mathbb{C}^{2}:\text{for every sequence }\{h_{k}\}\subset\mathcal{S}\text{ the sequence }\{h_{k}^{-1}(z)\}\text{ is bounded}\big{\}}.$

  2. (2)

     The weak positive filled Julia set is defined as the collection of all the points $z\in\mathbb{C}^{2}$ such that there exists a sequence $\{h_{k}\}\subset\mathcal{S}$ with $h_{k}\in\mathscr{G}_{n_{k}}$, where $n_{k}\to\infty$ as $k\to\infty$, and $\{h_{k}(z)\}$ is bounded. Similarly we define the weak negative filled Julia set, i.e.,

     $\displaystyle\mathbf{K}^{+}_{\mathcal{S}}=\big{\{}z\in\mathbb{C}^{2}:\text{ there exist }h_{k}\in\mathscr{G}_{n_{k}}\text{ such that }n_{k}\to\infty\text{ and }\{h_{k}(z)\}\text{ is bounded}\big{\}},$

     $\displaystyle\mathbf{K}^{-}_{\mathcal{S}}=\big{\{}z\in\mathbb{C}^{2}:\text{ there exist }\tilde{h}_{k}\in\mathscr{G}_{n_{k}}\text{ such that }n_{k}\to\infty\text{ and }\{\tilde{h}_{k}^{-1}(z)\}\text{ is bounded}\big{\}}.$

  Note that $\mathcal{K}_{\mathcal{S}}^{\pm}\subset\mathbf{K}_{\mathcal{S}}^{\pm}$ and these sets are uniquely associated to the semigroup $\mathcal{S}$.

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  Similarly as above we introduce the weak and strong escaping sets $\mathcal{U}^{\pm}_{\mathcal{S}}$ and $\mathbf{U}_{\mathcal{S}}^{\pm}$

  $$\mathcal{U}_{\mathcal{S}}^{\pm}=\mathbb{C}^{2}\setminus\mathcal{K}_{\mathcal{S}}^{\pm}\text{ and }\mathbf{U}_{\mathcal{S}}^{\pm}=\mathbb{C}^{2}\setminus\mathbf{K}_{\mathcal{S}}^{\pm}.$$

  Note that $\mathbf{U}_{\mathcal{S}}^{+}$ is the Fatou component at infinity with respect to the dynamics of the semigroup $\mathcal{S}.$ Similarly $\mathbf{U}_{\mathcal{S}}^{-}$ is the Fatou component at infinity for $\mathcal{S}^{-}$.

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  Finally, we define the cumulative positive and negative Julia sets for the semigroup $\mathcal{S}$, i.e.,

  $$\mathbf{J}_{\mathcal{S}}^{+}=\overline{\bigcup_{h\in\mathcal{S}}\mathcal{J}_{h}^{+}},\,\mathbf{J}_{\mathcal{S}}^{-}=\overline{\bigcup_{h\in\mathcal{S}}\mathcal{J}_{h}^{-}}.$$

Observe that, either of the sets $\mathcal{K}_{\mathcal{S}}^{+}$ or $\mathcal{K}_{\mathcal{S}}^{-}$ or both may be empty for some semigroups $\mathcal{S}$, of form (1.1). However, this situation does not affect the dynamics as such.

We will now explore some important properties of the sets introduced above via the filtration properties of the elements of $\mathcal{S}$, on appropriate domains. To discuss this in detail, let us first recall the definition of the sets $V_{R}$ and $V_{R}^{\pm}$ for some $R>0$, introduced in [21] (or [4]) for filtering the dynamics of (finite) compositions of generalised Hénon maps. They are $V_{R}=\big{\{}(x,y)\in\mathbb{C}^{2}:\max\{|x|,|y|\}\leq R\big{\}}$, the polydisk of radius $R$ and

Also recall the subsets $\mathscr{G}_{k}$ of $\mathcal{S}=\langle\mathscr{G}\rangle$, for every $k\geq 1$, defined as

We first record the dynamical behaviour of the semigroup $\mathcal{S}$ on $V_{R}^{\pm}$ for an appropriate $R>0$.

The constant $R_{\mathcal{S}}>0$ obtained in Lemma 2.3 is actually independent of the generators and will be referred along, as the radius of filtration for the semigroup $\mathcal{S}.$

Further, from the proof of Proposition 2.5 we get

The shaded region in Figure 1 corresponds to the weak filled Julia sets $\mathbf{K}_{\mathcal{S}}^{\pm}$, with the lighter shade representing the Fatou components and the darker shades representing the Julia sets $\mathcal{J}_{\mathcal{S}}^{\pm}$ contained in them, respectively.

Finally we conclude this section, by observing the following crucial fact which will be very important for further computations

In this section, we will first complete the proof of Theorem 1.3 and consequently observe a few important corollaries. To begin, let us recall the definition of the sequence of plurisubharmonic functions $\{G_{k}\}$, introduced in Section 1,

where $D=D_{\mathscr{G}}$ is the total degree corresponding to the generating set $\mathscr{G}$ of the semigroup $\mathcal{S}.$ First, we note the following straightforward consequence from the results in Section 2.

Now we are ready to present the proof of Theorem 1.3, which involves some steps.