A new proof of a theorem of Hubbard and Oberste-Vorth
- Remus Radu^{1}Email author and
- Raluca Tanase^{1}
https://doi.org/10.1186/s13663-016-0528-1
© Radu and Tanase 2016
Received: 10 November 2015
Accepted: 9 March 2016
Published: 31 March 2016
Abstract
We give a new proof of a theorem of Hubbard and Oberste-Vorth (Real and Complex Dynamical Systems, pp. 89-132, 1995) for Hénon maps that are perturbations of a hyperbolic polynomial and obtain the Julia set \(J^{+}\) inside a polydisk as the image of the fixed point of a contracting operator. We also give different characterizations of the Julia sets J and \(J^{+}\) which prove useful for later applications.
Keywords
MSC
1 Introduction
Fixed point theorems have found a lot of applications in dynamical systems in higher dimensions. They are used for proving the existence of the stable and unstable manifolds of a hyperbolic fixed point, or the existence of local foliations in the presence of a dominated splitting of the tangent bundle over an invariant set of a \(\mathcal{C}^{k}\) self-map of a Riemannian manifold. In this article we give a description of the global structure of the Julia sets J and \(J^{+}\) of a dissipative hyperbolic Hénon map in \(\mathbb{C}^{2}\) as the unique fixed point of a contracting operator in an appropriate function space. This provides an alternative proof of a well-known theorem of Hubbard and Oberste-Vorth [1], which was one of the starting points (along with [2, 3] and the works of Friedland and Milnor [4], Bedford and Smillie [5–7], Fornæss and Sibony [8], etc.) of more than two decades of research in dynamics in several complex variables. The proof that we give strengthens slightly the result of the theorem, and some of the tools developed here have found further applications to the study of Hénon maps with a semi-parabolic fixed point or cycle [9] and their perturbations [10].
Hubbard and Oberste-Vorth [1] studied the structure of the Julia sets J, \(J^{+}\), and \(J^{-}\) for Hénon maps which are small perturbations of a hyperbolic polynomial p. Polynomials and Hénon maps have some fundamental differences: polynomials are not injective whereas Hénon maps are, polynomials and their rate of escape functions have finitely many critical points, on the other hand Hénon maps do not have any critical points in the usual sense, but their associated rate of escape functions have infinitely many critical points. Starting from the polynomial p, Hubbard and Oberste-Vorth created some objects that carry bijective dynamics (projective and inductive limits), and used those to describe the dynamics of the Hénon map on its Julia sets (see Theorem 1.4 in [1]). Their proof relies on telescopes for hyperbolic polynomials and crossed mappings. We will give a new proof of the theorem for the sets J and \(J^{+}\) in the language of a fixed point theorem. We will recover the set \(J^{+}\) inside the bidisk \(\mathbb{D}_{r}\times\mathbb{D}_{r}\) as the image of the unique fixed point of a contracting graph-transform operator in some function space \(\mathcal{F}\), which we define in Section 4. We will complete the proof of the theorem in Section 5, when we establish conjugacies between the Hénon map and certain model maps. We also obtain other new characterizations of the Julia sets J and \(J^{+}\). The construction resembles the proof of the Hadamard-Perron theorem (see e.g. [11]). This approach has the advantage that it can be generalized to complex Hénon maps with a semi-parabolic fixed point [9], but the analysis in that case is much more complex (due to the loss of hyperbolicity) and requires several delicate arguments.
2 Tools from one-dimensional dynamics
An external ray \(R_{t}\) is the image under the Riemann mapping \(\psi _{p}\) of the straight line \(\{re^{2\pi i t}, r>1\}\). The Carathéodory loop is defined as \(\gamma(t)=\lim_{r\searrow1}\psi_{p}( re^{2\pi i t})\) and we say that the ray \(R_{t}\) lands at a point \(\gamma(t)\in J_{p}\) if this limit exists. The external ray \(R_{0}\) lands at the β-fixed point of p. An equipotential for the polynomial p is the image under the Riemann mapping \(\psi_{p}\) of the circle \(\{re^{2\pi i t}, t\in\mathbb{R}/\mathbb{Z}\}\) of radius \(r>1\).
A point x is called a critical point of p if \(p'(x)=0\), in which case \(c=p(x)\) is called a critical value. We say that p is hyperbolic if \(p'\) is expanding on a neighborhood of the Julia set.
Throughout this paper we assume that p is hyperbolic and has connected Julia set. In this case, the filled Julia set \(K_{p}\) is connected and locally connected, and none of the critical points of p belong to the Julia set \(J_{p}\) [13]. Moreover, all critical points of p are attracted to attracting cycles, and the number of attracting cycles is bounded above by \(d-1\), by the Fatou-Shishikura inequality. For each attracting cycle, we consider the union \(V_{i}\) of sufficiently small disks centered around the points of the cycle, such that \(V_{i}\) is contained in the immediate basin of attraction and \(p(V_{i})\) is relatively compact in \(V_{i}\). Set \(\Delta= \bigcup_{i=1}^{k} V_{i}\), where k is the number of attracting cycles. There exists a minimal iterate \(n\geq0\) such that \(p^{-\circ n}(\Delta)\) contains all critical values of p. So \(p^{-\circ(n+1)}(\Delta)\) belongs to the interior of the filled Julia set \(K_{p}\) and contains all critical points of p.
Since the Julia set \(J_{p}\) is locally connected, the sequence of equipotentials \(\gamma_{n}\) converges in the Poincaré metric of the set U to the Carathéodory loop γ of the polynomial p.
Lemma 2.1
Let z be a point in \(U'\) and let δ be small enough so that \(z-\delta\) is also a point in \(U'\). Then \(|\rho_{U}(z)-\rho_{U}(z-\delta)|\leq|\delta| C \rho_{U}(z)\).
The proof of the lemma is immediate and is left to the reader.
Lemma 2.2
Let \(z_{1}\) and \(z_{2}\) be any two points in \(U'\), and let δ be small enough so that \(z_{1}-\delta\) and \(z_{2}-\delta\) are still in \(U'\). Then \(d_{U}(z_{1}-\delta, z_{2}-\delta)\leq(1+|\delta| C) d_{U}(z_{1},z_{2})\).
Proof
3 Construction of the neighborhood V
Throughout this paper we will interchangeably use H and \(H_{p,a}\) to denote the Hénon map.
- (i)
\(\overline{H(V)}\) does not intersect the horizontal boundary of V, that is, \(|ax|< r\) for any \(x\in U'\).
- (ii)
\(J\subset V\). One can choose for instance \(r>3\) so that \(J\subset\mathbb{D}_{r}\times \mathbb{D}_{r}\) as above. Notice that \(J \cap\mathbb{D}_{r}\times\mathbb{D}_{r} = J \cap V\), by construction.
- (iii)
All points in \(H(V)-\mathbb{D}_{r}\times\mathbb{D}_{r}\) belong to the escaping set \(U^{+}\). One can choose R sufficiently large in equation (1) so that the circle \(\partial\mathbb{D}_{r}\) is contained in the set \(U' \). By part (i), any point in V that does not remain in \(\mathbb{D}_{r}\times \mathbb{D}_{r}\) under forward iteration of H belongs to the set \(W^{+}\), which is contained in \(U^{+}\).
- (1)
\(r|a|<\inf_{x\in U'}|p'(x)|\).
- (2)
\(r|a|<\operatorname{dist}(\partial U', \partial U)\). In other words, the \(r|a|\)-neighborhood of \(U'\) is compactly contained in U.
Lemma 3.1
Let \((x,y)\in V\) and \((x',y')=H^{-1}(x,y)\). If \(|y'|< r\) then \((x',y')\in V\).
Proof
The point \((x',y' )\) belongs to V iff \(x'=y/a\in U'\) and \(|y'|=|(x-p(y/a))/a|< r\). By hypothesis we have \(|x-p(y/a)|< r|a|\). The point x belongs to \(U'\) and \(|a|\) is chosen small enough so that the disk of radius \(r|a|\) around x is in U. It follows that \(p(x')\in U\), hence \(x'\in U'\). Therefore \((x',y')\) belongs to V. □
Proposition 3.2
- (a)
If \(|\xi'|<|\eta'|\) then \(|\xi|<|\eta|\).
- (b)
If \(|\xi|>|\eta|\) then \(|\xi'|>|\eta'|\).
Proof
- (a)
If \(|\xi'|<|\eta'|\) then \(|p'(x)||\xi|-|a||\eta|<|\xi '|<|\eta'|=|a||\xi|\), so \(|\xi|(|p'(x)|-|a|)<|a||\eta|\). The point \((x,y)\) belongs to V, so x is bounded away from the critical points of p, in fact we have \(|p'(x)|>r|a|\) where \(r>2\). Thus we get \(|\xi |<|\eta|\).
- (b)
If \(|\xi|>|\eta|\) then \(|\xi'|>|p'(x)|\xi|-|a||\eta |>(|p'(x)|-|a|)|\xi|>|a||\xi|=|\eta'|\).
Definition 3.3
Let \(\beta=\{(f(z),z),\, z\in\mathbb{D}_{r}\}\subset V\) be the graph of a holomorphic function \(f:\mathbb{D}_{r}\rightarrow U'\). We say that β is a vertical-like disk if for all points \((x,y)\) on β, the tangent vectors to β at \((x,y)\) belong to the vertical cone \(\mathcal{C}^{v}_{(x,y)}\).
Lemma 3.4
If β is a vertical-like curve in V then \(H^{-1}(\mathcal{\beta})\cap V\) is the union of d vertical-like curves.
Proof
Therefore the degree of the projection of \(H^{-1}(\beta)\) on the second coordinate is constant in \(\mathbb{C}\times\mathbb{D}_{r}\). It is easy to see that the degree of the projections is equal to the degree of the polynomial p, by looking at the number of intersections of \(H^{-1}(\beta)\) with the x-axis. The curve \(H(x\mbox{-axis})=\{ (p(x),ax),x\in\mathbb{C}\}\) has d connected components inside V, all horizontal-like. The curve β is a vertical-like disk in V, hence β intersects \(H(x\mbox{-axis})\) in exactly d points, which implies that \(H^{-1}(\beta)\) intersects the x-axis in d points.
Thus \(H^{-1}(\beta)\cap\mathbb{C}\times\mathbb{D}_{r}\) is a union of d analytic curves \(\beta_{i}\), \(i=0,1,\ldots, d-1\), which are all contained in V, by Lemma 3.1. The map \(pr_{2}: \beta_{i}\rightarrow\mathbb{D}_{r}\), \(pr_{2}(x,y)=y\) is a covering map of degree one. By the Inverse Function Theorem, \(\beta _{i}\) is the graph of a holomorphic function \(x=\phi(y)\) where \(\phi :\mathbb{D}_{r}\rightarrow U'\). The map ϕ must also be injective, because \(pr_{1}: \beta_{i}\rightarrow U'\), \(pr_{1}(x,y)=x\) is injective. By the Schwarz-Pick lemma, \(\phi:\mathbb{D}_{r}\rightarrow U'\) is weakly contracting in the Poincaré metrics of \(\mathbb{D}_{r}\) and \(U'\), hence strongly contracting if we endow \(U'\) with the Poincaré metric of U. By Lemma 3.2 we have \(|\phi'(z)|<1\) for \(z\in\mathbb{D}_{r}\). It follows that \(\beta_{i}\) is vertical-like. □
4 A fixed point theorem
Let \(\gamma_{0}\) be the equipotential of the polynomial p (see equation (2)) that defines the outer boundary of the set \(U'\).
Definition 4.1
We denote by \(f_{0}:\mathbb{S}^{1}\times\mathbb {D}_{r}\rightarrow V\) the map \(f_{0}(t,z)=(\gamma_{0}(t),z)\). The image of the map \(f_{0}\) is a solid torus which represents the outer boundary of the set V.
For any fixed \(t\in\mathbb{S}^{1}\), \(f_{0}(dt\times\mathbb{D}_{r})\) is a vertical disk in V, so \(H^{-1}\circ f_{0}(dt\times\mathbb{D}_{r})\cap V\) is a union of d vertical-like disks, by Lemma 3.4. Let \(C_{t}\) be the connected component of \(H^{-1}\circ f_{0}(dt\times \mathbb{D} _{r})\cap V\) that crosses the x-axis at \((\gamma _{1}(t),0 )\). Recall that \(\gamma_{1}\) is the equipotential of the polynomial p given by \(\gamma_{1}(t)=p^{-1}(\gamma_{0}(dt))\), where the choice of the appropriate inverse branch of p is made as in equation (2). Notice that \(pr_{2}:C_{t}\rightarrow\mathbb{D} _{r}\), \(pr_{2}(x,z)=z\) is a degree one covering map, hence \(C_{t}\) is the graph of a holomorphic function \(x=\varphi^{1}_{t}(z)\). This enables us to define a new function \(f_{1}:\mathbb{S}^{1}\times\mathbb {D}_{r} \rightarrow V\) as \(f_{1}(t,z)=(\varphi^{1}_{t}(z),z)\). Notice that \(f_{1}\) is homotopic to \(f_{0}\) by construction since \(\gamma_{1}\) and \(\gamma_{0}\) are homotopic. Moreover, since \(|a|\) is small, \(f_{1}(\mathbb {S}^{1}\times\mathbb{D} _{r})\) and \(f_{0}(\mathbb{S}^{1}\times\mathbb{D}_{r})\) are disjoint. Let \(\tilde {\delta}=d(f_{1}, f_{0})>0\). Notice that when \(|a|\) is small δ̃ is essentially the distance between \(\partial U'\) and \(\partial{U''}\), where \(U''=p^{-1}(U')\Subset U'\).
Let now \(R_{0}:[0,1]\times\mathbb{D}_{r}\rightarrow V\), \(R_{0}(0,z)=f_{0}(0,z)\), \(R_{0}(1,z)=f_{1}(0,z)\) be a homotopy of vertical-like disks connecting \(f_{0}(0\times\mathbb{D}_{r})\) to \(f_{1}(0\times\mathbb{D}_{r})\), such that \(R_{0}(s,0)\) is a point on the external ray of angle 0 of the polynomial p which connects \(\gamma_{0}(0)\) to \(\gamma_{1}(0)\). As before, \(H^{-1}(\operatorname{Im}(R_{0}))\cap V\) has d connected components. Denote by \(R_{1}\) the component that contains \(f_{1}(0\times\mathbb{D}_{r})\); \(R_{1}\) is a collection of vertical-like disks that can be parametrized as graphs over the second coordinate, \(R_{1}(s,z)=(\phi^{1}_{s}(z),z)\) for all \(s\in[0,1]\). Inductively, we can construct a sequence of (approximative) external ray segments \(R_{n}(s,z)=(\phi^{n}_{s}(z),z)\) by choosing the component of \(H^{-1}(\operatorname{Im}(R_{n-1}))\cap V\) that has the appropriate ‘matching end’, i.e. for which \(\phi^{n}_{0}(z)=\phi^{n-1}_{1}(z)\). The set \(\mathcal{R}=\bigcup_{n\geq0} R_{n}\) will be our approximation for the external 3-D ray of angle 0 for the Hénon map inside the set V.
Definition 4.2
Proposition 4.3
The map \(F:\mathcal{F'}\rightarrow\mathcal{F'}\) is well defined.
Proof
Theorem 4.4
The map \(F:\mathcal{F'}\rightarrow\mathcal{F'}\) is a contraction in the metric defined on \(\mathcal{F}\) and has an unique fixed point \(f^{*}\).
Proof
The following propositions describe the properties of the fixed point \(f^{*}\).
Proposition 4.5
For any fixed \(t\in\mathbb{S}^{1}\), \(f^{*}(t,z)=(\varphi_{t}(z),z)\), where the map \(\varphi_{t}:\mathbb{D}_{r}\rightarrow U'\) is holomorphic, and either injective or constant.
Proof
The fixed point \(f^{*}\) is obtained via the Banach fixed point theorem as the limit of the sequence \(f_{n}(t,z) = F^{\circ n}(f_{0})(t,z)\) for \(n\geq1\) and \(f_{0}(t,z)=(\gamma_{0}(t),z)\). By the construction of the function space, we can write \(f_{n}(t,z) = (\varphi^{n}_{t}(z),z)\), where \(\varphi ^{n}_{t}:\mathbb{D}_{r}\rightarrow U'\) are holomorphic and injective for \(n\geq1\). By Hurwitz’s theorem a uniform limit of holomorphic injective mappings is holomorphic and either injective or constant. □
Proposition 4.6
The function \(f^{*}:\mathbb{S}^{1}\times\mathbb{D}_{r}\rightarrow V\) is continuous with respect to \(t\in\mathbb{S}^{1}\), holomorphic with respect to \(z\in\mathbb{D} _{r}\) and holomorphic with respect to the parameter a.
Proof
As observed in the previous proposition, the map \(f^{*}\) is obtained as a uniform limit of the sequence \(f_{n}(t,z) = (\varphi ^{n}_{t}(z),z)\), where \(\varphi^{n}_{t}(z)\) is continuous in t and holomorphic in z. Thus \(f^{*}\) is continuous in t and holomorphic in z.
Clearly \(f_{0}(t,z)=(\gamma_{0}(t),z)\) does not depend on the parameter a. When \(|a|\) is small, each function \(f_{n}\) depends holomorphically on a. The construction of the metric space is uniform in a and so the limit \(f^{*}\) is holomorphic with respect to a. □
We can now recover the Julia set \(J^{+}\cap V\) as the image of the fixed point \(f^{*}\).
Lemma 4.7
\(J^{+}\cap V=\bigcap_{n\geq0}H^{-\circ n}(V)\).
Proof
Let q be any point in \(\bigcap_{n\geq0}H^{-\circ n}(V)\). Since all forward iterates of q remain in the bounded set V, q cannot belong to the escaping set \(U^{+}\). When H is hyperbolic, the interior of \(K^{+}\) consists of the basins of attraction of attractive periodic orbits [5]. However, the set \(U'\) does not contain any attractive cycles of the polynomial p so the set \(V=U'\times\mathbb{D}_{r}\) does not contain any attractive cycles of the Hénon map H for small values of the Jacobian. Since all forward iterates of q remain in V, q cannot belong to the interior of \(K^{+}\). Hence \(q\in J^{+}\).
Let now q be any point in \(J^{+}\cap V\). The Julia set J is contained in V. When H is hyperbolic, the Julia set \(J^{+}\) is the stable set of J, that is, \(W^{s}(J)=J^{+}\) [5]. It follows that q must belong to the stable manifold \(W^{s}(y)\) of some point \(y\in J\). So all forward iterates of q converge to the orbit of y, which is contained in J, hence also in V. In particular no forward iterate of q can exit V, hence \(q\in\bigcap_{n\geq0}H^{-\circ n}(V)\). □
Lemma 4.8
\(\operatorname{Im}(f^{*}) = J^{+}\cap V\).
Proof
By Lemma 4.7 we have \(J^{+}\cap V=\bigcap_{n\geq0}H^{-\circ n}(V) = \bigcap_{n\geq 0}H^{-\circ n}(V\cap\overline{U}^{+} )\). The set \(J^{+}\) is the topological boundary of the escaping set \(U^{+}\) and \(J^{+} \cap V\) is the inner boundary of the set \(V\cap\overline{U}^{+}\). Recall that \(f_{0}(t,z)=(\gamma_{0}(t),z)\). By construction, \(\operatorname{Im}(f_{0})\) is the outer boundary of V and is entirely contained in \(U^{+}\). Moreover, the sequence \(f_{n}:\mathbb{S}^{1}\times\mathbb {D}_{r}\rightarrow V\), \(f_{n}=F^{n}(f_{0})\) converges to the unique fixed point \(f^{*}\). The image \(f_{n}(\mathbb{S}^{1}\times\mathbb{D}_{r})\) is the outer boundary of the set \(\bigcap_{0\leq k\leq n}H^{-\circ k}(V\cap\overline{U}^{+})\). Hence \(\operatorname{Im}(f^{*})=\bigcap_{n\geq0}H^{-\circ n}(V\cap\overline{U}^{+})\). □
5 Characterizations of J and \(J^{+}\)
Theorem 5.1
Proof
The Julia set J is the set of points from \(J^{+}\) that do not escape to infinity under backward iterations of the Hénon map. By assumption (ii) from the construction of the neighborhood V in Section 3, \(J\subset V\). Thus \(J = \bigcap_{n\geq0}H^{\circ n}(J^{+}\cap V)\).
Let \(\Sigma^{+} = \bigcap_{n\geq0}\sigma^{\circ n}(\mathbb {S}^{1}\times\mathbb{D} _{r})\). The following is a direct consequence of Theorem 5.1 and the discussion above.
Theorem 5.2
The Julia set J of the Hénon map is homeomorphic to the quotient of the solenoid \(\Sigma^{+}\) by the equivalence relation ∼, which is further homeomorphic to the set \(\bigcap_{n\geq0}\psi^{\circ n}(J_{p}\times\mathbb{D}_{r})\).
Define the (inductive limit) space \(\check {J}_{p}\) as the quotient \((J_{p}\times\mathbb{D}_{r})\times\mathbb{N}/ \sim\), where the equivalence relation this time is defined by \((x,n)\sim(\psi (x),n+1)\). The space \(\check {J}_{p}\) comes with a natural bijective map \(\check {\psi}: \check {J}_{p}\rightarrow \check {J}_{p}\) given by \((x,n)\mapsto(\psi(x),n)\). One should think of the space \(\check {J}_{p}\) as an increasing union of sets homeomorphic to \(J_{p}\times\mathbb{D}_{r}\).
Theorem HOV
(Hubbard, Oberste-Vorth [1])
Proof
The map ψ can be conjugate to a map \(\psi'\) defined by \(\psi '(\zeta, z)= (p(\zeta),\zeta-\frac{\epsilon^{2}z}{p'(\zeta )} )\) by the linear change of variables \((\zeta,z)\mapsto(\zeta ,\epsilon z)\). The map \(\psi'\) is in fact the model map used in [1]. The difference comes from the fact that we are using the Hénon map normalized so that it has Jacobian \(-a^{2}\) rather than a.
The following theorem gives another perspective on the Julia set \(J^{+}\), without using the inductive limit space \(\check {J}_{p}\) (see also Theorem 5.2 in [16] and [17] for other characterizations).
Theorem 5.3
Proof
Let \((\zeta, z)\in J_{p}\times\mathbb{C}\) and let n be the first iterate such that \(\psi^{\circ n}(\zeta, z)\) belongs to \(J_{p}\times \mathbb{D}_{r}\). We define \(\Psi(\zeta,z) = H_{p,a}^{-\circ n}\circ\Phi^{+}\circ \psi^{\circ n}(\zeta, z)\), where \(\Phi^{+}\) is the conjugating homeomorphism from diagram (10). It is easy to check that the map Ψ is a surjective semi-conjugacy. The map Ψ is injective on \(J_{p}\times\mathbb{D}_{r}\), but not on \(J_{p}\times \mathbb{C}\). □
Declarations
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Authors’ Affiliations
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