A polynomial map $ f : \mathbb{C}^2\to\mathbb{C}^2$ is called a polynomial diffeomorphim of $ \mathbb{C}^2$ if it has a polynomial inverse. Examples of polynomial diffeomorphims of $ \mathbb{C}^2$ are an affine map : \begin{eqnarray*} \alpha : (x, y)\longmapsto (a_1 x+b_1 y+c_1, a_2 x+b_2 y+c_2) \end{eqnarray*} where $ a_1b_2-a_2b_1\ne 0$, an elementary map : \begin{eqnarray*} \beta : (x, y)\longmapsto (ax+c, p(x)+by) \end{eqnarray*} where $ p(x)$ is a polynomial of degree $ d\geq 2$ and $ ab\ne 0$, and a generalized Hénon map : \begin{eqnarray*} f_{p, b} : (x, y)\longmapsto (p(x)-by, x) \end{eqnarray*} where $ p(x)$ is a polynomial of degree $ d\geq 2$ and $ b\in\mathbb{C}^{\times}\equiv\mathbb{C}{\setminus} \{0\}$.

Let $ \mathrm{Poly}(\mathbb{C}^2)$ be the space of polynomial diffeomorphisms of $ \mathbb{C}^2$. This forms a group by the composition of two maps and the conjugacy classes can be classified into three types.

The proof of Theorem 2.1 is based on a classical result of [Jung1942] which claims that the group $ \mathrm{Poly}(\mathbb{C}^2)$ is generated by the affine maps and the elementary maps. One may wonder if an analogous result holds for the group of polynomial diffeomorphisms of $ \mathbb{C}^3$. In his 1972 paper ([Nagata1972]), Masayoshi Nagata He was nicknamed "Mr. Counterexample" with admiration. He also gave a counterexample to the 14th problem of David Hilbert. See the paper by M. Miyanishi, Masayoshi Nagata (1927–2008) and his mathematics , Kyoto J. Math. 50 (2010), no. 4, 645–659.   × ⁴ proposed the map: \begin{eqnarray*} (x, y, z)\longmapsto (x+(x^2-yz)z, y+2(x^2-yz)x+(x^2-yz)^2z, z) \end{eqnarray*} as a possible counterexample to this analogy. More than 30 years later, [Shestakov and Umirbaev2004] finally showed Nagata's conjecture in the affirmative, i.e. the Nagata map is not contained in the subgroup of $ \mathrm{Poly}(\mathbb{C}^3)$ generated by affine maps and the elementary maps.

It is easy to see that the dynamics of the cases (a) and (b) in Theorem 2.1 is simple, so the only dynamically interesting case is (c). Therefore, we will hereafter treat a map of the form: \begin{eqnarray*} f=f_{p_1, b_1}\circ \cdots\circ f_{p_k, b_k}. \end{eqnarray*}

Note that for a map of this form, the Jacobian determinant is given by $ \det (Df)=b_1\cdots b_k$. We define $ d\equiv d_1\cdots d_k$ and call it the degree of $ f$, where $ d_i\equiv \deg p_i$. The next result indicates that the maps in this class exhibit rich dynamics.

Let us define the forward/backward filled-Julia sets of $ f$ as \begin{eqnarray*} K^{\pm}\equiv \big\{(x, y)\in\mathbb{C}^2 : \{f^{\pm n}(x, y)\}_{n\geq 0} \mbox{ is bounded} \big\}, \end{eqnarray*} the forward/backward Julia sets of $ f$ as $ J^{\pm}\equiv \partial K^{\pm}$. We also define $ K_f\equiv K^+\cap K^-$. We put $ J=J_f\equiv J^+\cap J^-$ and call it the Julia set We are also interested in the set $ J_f^{\ast}$ defined as the support of the unique maximal entropy measure ([Bedford and Smillie1991a], [Bedford et al.1993a]) (see Sect. 3.1). We see that $ J_f^{\ast}\subset J_f$ holds in general, and $ J_f^{\ast}=J_f$ can be shown when $ f$ is hyperbolic ([Bedford and Smillie1991a]).   × ⁵ of $ f$.

As a comparison with the quadratic family $ p_c$ we consider the complex Hénon family : \begin{eqnarray*} f_{c, b} : (x, y)\longmapsto (x^2+c-by, x) \end{eqnarray*} defined on $ \mathbb{C}^2$, where $ (c, b)\in\mathbb{C}\times\mathbb{C}^{\times}$ is a parameter. Let us call $ f_{c, b}$ real if $ (c, b)\in\mathbb{R}\times\mathbb{R}^{\times}$. When $ f_{c, b}$ is real, the dynamical system $ f_{c, b} : \mathbb{R}^2\to\mathbb{R}^2$ is well-defined.

Based on the ten items discussed in Sect. 1, we propose the following ten problems for polynomial diffeomorphisms $ f$ of $ \mathbb{C}^2$ or the Hénon family.

Problems:

1. (i) Define "dynamical critical points" of $ f$. Related it to the connectivity of the Julia set.
2. (ii) When the Julia set is connected, define the notion of external rays.
3. (iii) Establish a criterion for hyperbolicity of $ f$ on the Julia set and construct examples.
4. (iv) When the Julia set is connected and hyperbolic, describe it as a quotient space of a "simple" space like a circle.
5. (v) Define the notion of a Hubbard tree for $ f$. Prove that it reconstructs the Julia set.
6. (vi) Construct a (tight) automaton for $ f$. Prove that it reconstructs the Julia set.
7. (vii) Define the notion of an iterated monodromy group for $ f$. Prove that its limit space is homeomorphic to the Julia set.
8. (viii) Study the monodromy representation for the complex Hénon family. Is it surjective?
9. (ix) Establish a dynamics-parameter correspondence for the complex Hénon family.
10. (x) Characterize the hyperbolic horseshoe locus and the maximal entropy locus for the real Hénon family.

In the rest of this article, we will discuss the above problems for polynomial diffeomorphisms of $ \mathbb{C}^2$ or for the Hénon family. Section 3 is devoted to Problems (i), (ii) and (iv) where the results are obtained in [Bedford and Smillie1998a], [Bedford and Smillie1998b], [Bedford and Smillie1999]. Section 4 is the "Intermezzo" of this article and noting to do with the problem list above, where we discuss an application of the convergence theorem of currents ([Bedford and Smillie1991a], [Bedford and Smillie1991b], [Bedford and Smillie1992]) to curious objects in general topology called the Lakes of Wada . Section 5 discusses Problem (iii) and we present a construction of a hyperbolic Hénon map with intrinsically two-dimensional dynamics in [Ishii2008]. The following Section 6 is dedicated to Problems (v)–(vii) where we present some results in [Ishii2009], [Ishii2014]. Problems (viii) and (ix) are discussed in Sect. 7 and two conjectures from [Lipa2009] are presented. Finally in Sect. 8 we consider Problem (x) and present some related results in [Bedford and Smillie2004], [Bedford and Smillie2006], [Arai and Ishii2015], [Arai et al.2017].

We first introduce the following notion.

The following fundamental theorem states that the connectivity of $ J_f$ can be detected through the complex one-dimensional slice of $ J_f$ by some/any unstable manifold.

Indeed, Bedford and Smillie showed (Theorem 0.1 in [Bedford and Smillie1998b]) that (2) and (3) above are equivalent without $ |\det(Df)| \leq 1$, and called a map $ f$ satisfying the conditions unstably connected .

Our next task is to restate the conditions (2) and (3) in the theorem above so that they can be verified by computer experiments. To do this, let us introduce the Green functions of $ f$ as \begin{eqnarray*} G^{\pm}(x, y)\equiv \lim_{n\to+\infty}\frac{1}{d^n}\log^+\|f^{\pm n}(x, y)\|. \end{eqnarray*} One can see that $ G^{\pm}(x, y)$ are continuous and plurisubharmonic and satisfy $ G^{\pm}(f(x, y))=d^{\pm 1}\cdot G^{\pm}(x, y)$ on $ \mathbb{C}^2$, pluriharmonic on $ \mathbb{C}^2{\setminus} K^{\pm}$ and $ G^{\pm}(x, y)> 0$ iff $ (x, y)\in \mathbb{C}^2{\setminus} K^{\pm}$. Therefore, $ \mu^{\pm}\equiv \frac{1}{2\pi}dd^cG^{\pm}$ define positive $ (1, 1)$-currents on $ \mathbb{C}^2$.

Define an analogy of the Böttcher coordinate: \begin{eqnarray*} \varphi^+(x, y)\equiv \lim_{n\to +\infty}(\pi_x\circ f^n(x, y))^{\frac{1}{d^n}} \end{eqnarray*} (by choosing an appropriate $ d^n$-th root) for $ (x, y)\in V_R^+\equiv \{(x, y)\in \mathbb{C}^2 : |x|> R, \, |x|> |y|\}$, where $ \pi_x$ is the projection to the $ x$-axis and $ R> 0$ large. Note that we have $ \varphi^+(x, y)/x\to 1$ as $ |x|\to\infty$ for every fixed $ y$ and $ G^+(x, y)=\log|\varphi^+(x, y)|$ for $ (x, y)\in V_R^+$.

It was observed in [Bedford and Smillie1998b] that, when we try to extend $ \varphi^+$ along $ J^-{\setminus} K^+$, an obstruction is the critical points of $ G^+$ on $ W^u(q){\setminus} K^+$. This leads to define the dynamical critical set as \begin{eqnarray*} \mathcal{C}^u\equiv \bigcup_{q\in\mathcal{R}}\mathrm{Crit}(G^+; q), \end{eqnarray*} where $ \mathcal{R}$ denotes the set of Pesin regular points in $ J_f$ (e.g. the saddle periodic points) and \begin{eqnarray*} \mathrm{Crit}(G^+; q)\equiv \big\{(x, y)\in W^u(q){\setminus} K^+ : (x, y) \mbox{ is a critical point of } G^+|_{W^u(q){\setminus} K^+} \big\} \end{eqnarray*} for $ q\in\mathcal{R}$ (see [Bedford and Smillie1998a]). These critical points represent tangencies between the lamination of $ J^-$ by unstable manifolds and the foliation on $ \mathbb{C}^2{\setminus} K^+$ defined by the holomorphic $ 1$-form $ \partial G^+$. By the laminar structure of $ \mu^-$, it induces a measure $ \mu^-_c$ on the dynamical critical set $ \mathcal{C}^u$. This measure yields a formula for the Lyapunov exponent of $ f$ ([Bedford and Smillie1998a]): \begin{eqnarray*} \Lambda_{\mu}(f)=\log d+\int_{\{1\leq G^+< d\}}G^+d\mu^-_c \end{eqnarray*} with respect to the unique maximal entropy measure $ \mu\equiv \mu^+\wedge\mu^-$ (see [Bedford et al.1993a] for more details), where $ \{1\leq G^+< d\}$ is a fundamental domain for $ \mathcal{C}^u$. This formula generalizes the corresponding one-dimensional formula given in [Przytycki1985] and was a key step in the proof of Theorem 3.2.

The following claim has been obtained as a combination of an argument by Dujardin and ones in [Bedford and Smillie1998b]. For the proof we refer to [Ishii2011].

In particular, Theorems 3.2 and 3.3 yield

This justifies the algorithm of the program SaddleDrop ([SaddleDrop2000]) to draw the connectedness locus of the complex Hénon family. SaddleDrop was written around 2000 by Karl Papadantonakis, then an undergraduate student at Cornell. The procedure to use SaddleDrop is as follows.

• Step 1: Choose $ (c_0, b_0)\in \mathbb{C}\times \mathbb{C}^{\times}$ with $ |b_0|\leq 1$.
• Step 2: Compute $ W^u(q)$ of a saddle fixed point $ q$ for $ f_{c_0, b_0}$.
• Step 3: Draw the set $ K^+\cap W^u(q)$ as well as some equi-potential curves of $ G^+|_{W^u(q)}$ for $ f_{c_0, b_0}$ in the uniformized coordinate $ \mathbb{C}\cong W^u(q)$.
• Step 4: By looking at equi-potential curves of $ G^+|_{W^u(q)}$ in $ \mathbb{C}\cong W^u(q)$, we try to find an element in $ \mathrm{Crit}(G^+; q)$.
• Step 5: If we can find a critical point, click it; then SaddleDrop traces all parameters $ (c, b_0)\in \mathbb{C}\times \mathbb{C}^{\times}$ to which a continuation of the chosen critical point survives. Repeat this procedure for as many points in $ \mathrm{Crit}(G^+; q)$ as you can find.
• Step 6: Choose a new $ c_0\in \mathbb{C}$ (by keeping $ b_0\in \mathbb{C}^{\times}$) which was not traced from any of the previous choices and return to Step 2.

1. (i) If $ (c, b_0)$ can be traced from some $ (c_0, b_0)$ through a critical point, then we are sure (up to numerical error) that the Julia set of $ f_{c, b_0}$ is disconnected.
2. (ii) If $ (c, b_0)$ can not be traced from any $ (c_0, b_0)$ through any critical points we found, then the Julia set of $ f_{c, b_0}$ is "presumably" connected.

The claim (ii) is valid at best "presumably" because we do not know if the intersection of the complement of the connectedness locus with the slice $ \{(c, b)\in \mathbb{C}\times\mathbb{C}^{\times} : b=b_0\}$ is connected and because it is not theoretically possible to do Step 5 in the algorithm for all points in $ \mathrm{Crit}(G^+; q)$ which could be an infinite set. Modulo these issues, it seems that SaddleDrop may give a good approximation of the connectedness locus for the complex Hénon family. We refer to [Koch2010] for several pictures obtained by SaddleDrop as well as some other issues related to the algorithm.

In this section we define the notion of external rays and discuss a topological model for a connected and hyperbolic Julia set. We say that a polynomial diffeomorphism $ f$ of $ \mathbb{C}^2$ is hyperbolic if its Julia set $ J_f$ is a hyperbolic set for $ f$.

Let $ p_d : \mathbb{C}{\setminus}\overline{\Delta}\to\mathbb{C}{\setminus}\overline{\Delta}$ be the monomial map $ p_d(z)\equiv z^d$ of degree $ d\geq 2$. The projective limit of $ p_d : \mathbb{C}{\setminus}\overline{\Delta}\to\mathbb{C}{\setminus}\overline{\Delta}$ denoted as $ \mathbb{S}_d^{\mathbb{C}}$ together with the shift map on it $ \hat{p}_d : \mathbb{S}_d^{\mathbb{C}}\to\mathbb{S}_d^{\mathbb{C}}$ is called the complex solenoid of degree $ d$ (denoted as $ \Sigma_+$ in [Bedford and Smillie1999]). Similarly, let $ \delta_d : \mathbb{T}\to\mathbb{T}$ be the map given by $ \delta_d(\theta)\equiv d\cdot\theta$. The projective limit of $ \delta_d : \mathbb{T}\to\mathbb{T}$ denoted as $ \mathbb{S}_d^{\mathbb{R}}$ together with the shift map on it $ \hat{\delta}_d : \mathbb{S}_d^{\mathbb{R}}\to\mathbb{S}_d^{\mathbb{R}}$ is called the real solenoid of degree $ d$ (denoted as $ \Sigma_0$ in [Bedford and Smillie1999]). In the context of complex Hénon maps, real solenoids had appeared earlier, although in quite a different form, in [Hubbard1986], [Hubbard and Oberste-Vorth1994] as "the dynamics at infinity".

The next theorem states that, when $ J_f$ is connected and $ f$ is hyperbolic, one can define the notion of external rays in $ J^-{\setminus} K^+$ which are parameterized by the "space of angles" $ \mathbb{S}_d^{\mathbb{R}}$.

Indeed, it is shown in [Bedford and Smillie1998b] that, if $ |\det(Df)| \leq 1$ and $ J_f$ is connected, the holomorphic function $ \varphi^+ : V_R^+\to \mathbb{C}{\setminus} \{|z|\leq R\}$ extends to $ \varphi^+ : J^-{\setminus} K^+\to \mathbb{C}{\setminus} \overline{\Delta}$. Hence one can define \begin{eqnarray*} \Phi \, : \, J^-{\setminus} K^+\ni (x, y)\longmapsto (\varphi^+\circ f^n(x, y))_{n\in\mathbb{Z}}\in\mathbb{S}_d^{\mathbb{C}}. \end{eqnarray*} When moreover $ f$ is hyperbolic, $ \Phi : J^-{\setminus} K^+\to\mathbb{S}_d^{\mathbb{C}}$ is a finite covering ([Bedford and Smillie1999]). By modifying the "local inverse map" of $ \Phi$ appropriately, we obtain the homeomorphism $ \Psi : \mathbb{S}_d^{\mathbb{C}} \to J^-{\setminus} K^+$ in Theorem 3.5 (see Section 4 in [Bedford and Smillie1999]). It is still an open question if $ \Phi$ itself is a homeomorphism (see a remark just after Corollary 4.2 of [Bedford and Smillie1999]). If it is the case, we have $ \Psi=\Phi^{-1}$.

Thanks to the theorem above one can define (when $ f$ is hyperbolic) the notion of external rays in $ J^-{\setminus} K^+$ as the push-forward of the rays in $ \mathbb{S}_d^{\mathbb{C}}$ by $ \Psi$. Hyperbolicity of $ f$ also implies that every external ray has a well-defined landing point in $ J_f$. In particular, we have

In particular, $ f : J_f\to J_f$ is conjugate to the factor $ \hat{\delta}_d/_{\sim_{\mathrm{BS}}} : \mathbb{S}_d^{\mathbb{R}}/_{\sim_{\mathrm{BS}}}\to \mathbb{S}_d^{\mathbb{R}}/_{\sim_{\mathrm{BS}}}$, where we define $ \underline{\theta}\sim_{\mathrm{BS}} \underline{\theta}'$ iff $ \Psi(\underline{\theta})=\Psi(\underline{\theta}')$ for $ \underline{\theta}, \underline{\theta}'\in \mathbb{S}_d^{\mathbb{R}}$. The nature of $ \sim_{\mathrm{BS}}$ is, however, still mysterious (cf. the thesis of [Oliva1998]) and we will discuss this issue in Sect. 6.

First we remark that as a consequence of the convergence theorem of currents, Bedford and Smillie obtained the following curious result.

This result reminds us the so-called Lakes of Wada ; mutually disjoint three domains in the plane which possess common boundary. Such a curious example was first constructed in an article of [Yoneyama1917]. Here we quote the following nice explanation from [Hubbard and Oberste-Vorth1995]: Consider a circular island, inhabited, to the sorrow of the others, by three philanthropists. One has a lake of water, another of milk and a third of wine. The first, in a fit of generosity, decides to built a network of canals bringing water within 100m of every spot of the island. It is clearly possible to do this keeping the union of the original water lake and the water canals connected and simply connected, with closures disjoint from the other lakes.

Next the second, perhaps worried about child nutrition, decides to bring milk to within 10m of every spot on the island, and builds a system of canals to that effect. She also keeps her milk locus connected and simply connected.

Not to be outdone, the purveyor of wine now decides to bring wine to within 1m of every spot on the island. He finds his canal building rather more of an effort than the previous two, but being properly fortified, he carries it out.

In turn, each of the three philanthropists brings his or her product closer to the poor inhabitants. It should be clear that the construction can be continued, and that in the limit the construction achieves the desired result: each of the lakes, being an increasing union of connected and simply connected open sets, is a connected, simply connected set, and each point of the boundary of one is in the boundary of the other two. In the same paper Yoneyama writes that the construction "was informed to me by Mr. Wada." (see the footnote in page 60 of [Yoneyama1917]). This is why such domains are now called Lakes of Wada. But, who is this Mr. Wada? Here is his picture (Fig. 8). By courtesy of Mathematics Library at Kyoto University.   × ⁶

He is Takeo Wada, a Japanese mathematician working on analysis and general topology. He was one of the first students in the mathematics department of Kyoto Imperial University (now Kyoto University) and he later became a full professor there. According to J. H. Przytycki, most likely Wada published the first paper in Japan devoted to topology in 1911/1912. See his article Notes to the early history of the knot theory in Japan , arXiv:math/0108072.   × ⁷

Figure 8 Takeo Wada (1882–1944) .

A Fatou–Bieberbach domain in $ \mathbb{C}^2$ is a proper subdomain in $ \mathbb{C}^2$ which is biholomorphically equivalent to $ \mathbb{C}^2$. Since any attractive basin of $ f$ is contained in $ K^+$ and since $ \mathbb{C}^2{\setminus} K^+$ is non-empty, it is always a Fatou–Bieberbach domain.

Another question related to Theorem 4.1 is the existence/non-existence of a Fatou–Bieberbach domains with smooth boundaries. There exists a Fatou–Bieberbach domain with $ C^{\infty}$-smooth boundary by using a non-autonomous iterations ([Stens[U+00F8]nes1997]). On the other hand, as a consequence of Theorem 4.1 we have

Therefore, the only remaining case where the boundary of an attractive basin could be smooth is when $ f$ has only one attractive basin. Very recently the following result has appeared.

This result gives a complete answer to the question mentioned above.

Let $ p : \mathbb{C}\to\mathbb{C}$ be a polynomial of $ \deg p\geq 2$ and let $ J_p$ is its Julia set. Following ([Hubbard and Oberste-Vorth1995]) we denote by $ \hat{J}_p\equiv \varprojlim (p, J_p)$ the projective limit of $ p : J_p\to J_p$ and by $ \hat{p} : \hat{J}_p\to \hat{J}_p$ the shift map on it.

As an example of hyperbolic polynomial diffeomorphisms of $ \mathbb{C}^2$, it is known that a small perturbation of an expanding polynomial is hyperbolic. More precisely,

Next we introduce a criterion for hyperbolicity of polynomial diffeomorphisms $ f$ of $ \mathbb{C}^2$. Let $ A_x$ and $ A_y$ be bounded domains in $ \mathbb{C}$ and let $ \mathcal{A}=A_x\times A_y$. We then have projections $ \pi_x : \mathcal{A}\to A_x$ and $ \pi_y : \mathcal{A}\to A_y$. The following condition has been first introduced in [Hubbard and Oberste-Vorth1995] when $ \mathcal{A}$ is a polydisk (see [Ishii2008], [Ishii and Smillie2010] for more general case).

Let $ \mathcal{F}_h=\{A_x(y)\}_{y\in A_y}$ be the horizontal foliation of $ \mathcal{A}\cap f^{-1}(\mathcal{A})$ with leaves $ A_x(y)=(A_x\times \{y\})\cap (\mathcal{A}\cap f^{-1}(\mathcal{A}))$, and $ \mathcal{F}_v=\{A_y(x)\}_{x\in A_x}$ be the vertical foliation of $ \mathcal{A}\cap f^{-1}(\mathcal{A})$ with leaves $ A_y(x)=(\{x\}\times A_y)\cap (\mathcal{A}\cap f^{-1}(\mathcal{A}))$. Another condition we employ is the following ([Ishii2008], [Ishii and Smillie2010]).

Let $ |\cdot |_{A_x}$ and $ |\cdot |_{A_y}$ be Poincaré metrics in $ A_x$ and $ A_y$ respectively. The horizontal Poincaré cone field $ (\{C^h_q\}_{q\in \mathcal{A}}, \| \cdot \|_h)$ is \begin{eqnarray*} C^h_q\equiv \bigl\{v=(v_x, v_y) \in T_q \mathcal{A} : |v_x|_{A_x}\geq |v_y|_{A_y} \bigr\} \end{eqnarray*} with the metric $ \|v\|_h\equiv |D\pi_x (v)|_{A_x}$. The vertical Poincaré cone field $ (\{C^v_q\}_{q\in \mathcal{A}}, \| \cdot \|_v)$ is \begin{eqnarray*} C^v_q\equiv \bigl\{v=(v_x, v_y) \in T_q \mathcal{A} : |v_x|_{A_x}\leq |v_y|_{A_y} \bigr\} \end{eqnarray*} with the metric $ \|v\|_v\equiv |D\pi_y (v)|_{A_y}$. A product set $ \mathcal{A}=A_x\times A_y$ equipped with the horizontal and the vertical Poincaré cone fields is called a Poincaré box .

A crossed mapping $ \iota_{\mathcal{A}}, f : \mathcal{A}\cap f^{-1}(\mathcal{A})\to \mathcal{A}$ is said to expand the horizontal Poincaré cone field if there exists $ \lambda> 1$ so that for any $ q\in \mathcal{A}\cap f^{-1}(\mathcal{A})$, we have $ D\iota_{\mathcal{A}}^{-1}(C^h_{\iota_{\mathcal{A}}(q)})\subset Df^{-1}(C^h_{f(q)})$ and $ \lambda\|D\iota_{\mathcal{A}} (v)\|_h\leq \|Df(v)\|_h$ for any $ v\in T_q(\mathcal{A}\cap f^{-1}(\mathcal{A}))$ with $ D\iota_{\mathcal{A}}(v)\in C^h_{\iota_{\mathcal{A}}(q)}$. Similarly, a crossed mapping is said to contract the vertical Poincaré cone field if there exists $ \lambda> 1$ so that for any $ q\in \mathcal{A}\cap f^{-1}(\mathcal{A})$, we have $ Df^{-1}(C^v_{f(q)})\subset D\iota_{\mathcal{A}}^{-1}(C^v_{\iota_{\mathcal{A}}(q)})$ and $ \lambda\|Df(v)\|_v\leq \|D\iota_{\mathcal{A}} (v)\|_v$ for any $ v\in T_q(\mathcal{A}\cap f^{-1}(\mathcal{A}))$ with $ Df(v)\in C^v_{f(q)}$. A crossed mapping is called a hyperbolic system if it expands the horizontal Poincaré cone field and contracts the vertical Poincaré cone field.

The following statements give hyperbolicity criterion for polynomial diffeomorphisms of $ \mathbb{C}^2$ with a single Poincaré box.

Note that there are more checkable criteria for a map to be a crossed mapping called the boundary compatibility condition (Definition 2.15 in [Ishii2008]) and for the non-tangency condition called the off-criticality condition (Definition 2.16 in [Ishii2008]) which can be verified by hand or by computer assistance. Together with the technique of homotopy shadowing developed in [Ishii and Smillie2010], one obtains a quantitative estimate for the constant $ b_{\ast}$ in Theorem 5.2 and a new proof of the latter half of the statement in the theorem.

We remark that Theorem 5.5 as well as Corollary 5.6 holds for a map $ f$ from a Poincaré box to a different Poincaré box (see Corollary 2.17 in [Ishii2008]). Moreover, one can extend them to the case where $ f$ is a system of maps from the disjoint union of finitely many Poincaré boxes to itself (see Corollary 2.18 in [Ishii2008]).

Figure 9 Poincaré boxes for the non-planar map in Theorem 5.8 .

Theorem 5.2 tells that the dynamics of a Hénon map obtained as a small perturbation of an expanding polynomial is intrinsically complex one-dimensional. Therefore, a natural question arise; is there a hyperbolic polynomial diffeomorphism which is non-planar? The first non-planar example of a hyperbolic Hénon map was constructed by the author ([Ishii2008]).

The proof of Theorem 5.8 goes as follows. First we choose four Poincaré boxes $ \{\mathcal{A}_i\}_{i=0}^3$ in $ \mathbb{C}^2$ where $ \mathcal{A}_i=A_{x, i}\times A_{y, i}$ (see Fig. 9). Note that $ A_{x, 0}$ and $ A_{x, 3}$ are annuli so that $ \mathcal{A}_0$ and $ \mathcal{A}_3$ have vertical holes which are shaded in Fig. 9. Also note that, $ \mathcal{A}_1$ and $ \mathcal{A}_2$ are drawn at the same place to simplify the figure, although they are actually disjoint. Set $ \Sigma^0\equiv \{0, 1, 2, 3\}$ and \begin{eqnarray*} \Sigma^1\equiv \big\{(0, 3), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2), (3, 0), (3, 1), (3, 2)\big\}. \end{eqnarray*}

We first show that the union $ \bigcup_{i\in\Sigma^0}\mathcal{A}_i$ covers the Julia set $ J_f$ of the non-planar map $ f=f_{a, b}$ in Theorem 5.8 with computer assistance. Next, the multivalued dynamical system:

We show that $ \iota_{\mathcal{A}}, f : \mathcal{A}_i\cap f^{-1}(\mathcal{A}_j)\to\mathcal{A}_j$ is a crossed mapping satisfying the non-tangency condition for all $ (i, j)\in \Sigma^1$ by verifying the checkable conditions mentioned above again by computer assistance. This implies that (5.2) is a hyperbolic system in an extended sense.

Note that since the boxes $ \{\mathcal{A}_i\}_{i=0}^3$ have overlaps in $ \mathbb{C}^2$, this does not immediately imply the hyperbolicity of the original system (5.1). To overcome this point, we define a new horizontal cone at a point in the overlap as \begin{eqnarray*} C_p^{\cap}\equiv \bigcap_{i\in I(p)}C^i_p \end{eqnarray*} for $ p\in \mathcal{A}^0$, where $ C^i_p$ is the horizontal Poincaré cone at $ p\in\mathcal{A}_i$ with respect to the Poincaré box $ \mathcal{A}_i$ and $ I(p)\equiv \left\{i\in \Sigma^0 : p\in \mathcal{A}_i\right\}$. We also define a metric $ \| \cdot \|_{\cap}$ in it by \begin{eqnarray*} \|v\|_{\cap}\equiv \min \big\{\|v\|_{\mathcal{A}_i} : i\in I(p) \big\} \end{eqnarray*} for $ v\in C_p^{\cap}$ (see Definition 4.1 in [Ishii and Smillie2010]). One can then verify that it is invariant and expanding (Proposition 4.3 in [Ishii and Smillie2010]) and indeed non-empty (Corollary 4.21 in [Ishii and Smillie2010]). This yields the hyperbolicity of the cubic Hénon map in Theorem 5.8 on its Julia set.

Next we prove the non-planarity of the cubic Hénon map $ g=f_{a, b}$. Suppose that $ g : J_g\to J_g$ is topologically conjugate to $ \hat{p} : \hat{J}_p\to \hat{J}_p$ for an expanding polynomial $ p$. Then, by the comparison of the entropy, we know that the degree of $ p$ is $ 3$. Since $ g$ has an attractive $ 2$-cycle and since its Julia set $ J_g$ is neither connected nor totally disjoint, we know that one of the two critical points of $ p$ escapes to infinity and the other converges to an attractive $ 2$-cycle. This puts certain constraints on $ \hat{J}_p$. On the other hand, by using the four Poincaré boxes in the proof of Theorem 5.8, we can analyze the topology of $ J_g$ in terms of a symbolic dynamics (Theorem 4.23 in [Ishii and Smillie2010]). By comparing the topological types of some path-components of $ J_g$ and $ \hat{J}_p$, we finally arrive at a contradiction (see the end of Section 4 [Ishii2008] for more details).

Let us first explain the construction of Hubbard trees ([Ishii2009]) by using the non-planar map obtained in Theorem 5.8. Let $ \{\mathcal{A}_i\}_{i=0}^3$ be the family of Poincaré boxes appeared in the proof of Theorem 5.8 where $ \mathcal{A}_i=A_{x, i}\times A_{y, i}$. Define the forward Julia set of the disjoint system (5.2) by \begin{eqnarray*} J_+^0(\tilde{f})\equiv \bigcap_{n\geq 0}(\tilde{f}\circ \tilde{\iota}_{\mathcal{A}}^{-1})^{-n}(\widetilde{\mathcal{A}}^0) \end{eqnarray*} (and similar definition for $ J_+^1(\tilde{f})$). Since $ A_{y, i}$ is simply connected and the disjoint system (5.2) is a hyperbolic system, one can show that $ J_+^m(\tilde{f})\cap \mathcal{A}_i$ forms a lamination where every leaf is a vertical disk of degree one in $ \mathcal{A}_i$, i.e. the projection of the disk to $ A_{y, i}$ is a proper map of degree one. Moreover, one may assume that these disks are straight vertical (see the comment following Lemma 5.5 of [Ishii and Smillie2010]). In particular, the image of any leaf of the lamination by the projection $ \pi_x : \mathcal{A}_i\to A_{x, i}$ is one point. Therefore, by letting $ \mathcal{S}_i\equiv A_{x, i}$, $ \tilde{\sigma}\equiv \pi_x\circ \tilde{f}\circ\pi_x^{-1}$ and $ \tilde{\iota}_{\mathcal{S}}\equiv \pi_x\circ \tilde{\iota}_{\mathcal{A}}\circ\pi_x^{-1}$, we obtain the multivalued dynamical system:

Since the disjoint system (5.2) expands the horizontal Poincaré cone field, the system (6.1) equipped with the Poincaré metrics in $ \widetilde{\mathcal{S}}^m$ ($ m=0, 1$) is expanding in the sense of Definition 1.8.

Now we proceed as explained in item (v) in Sect. 1, but here is a crucial difference which comes from the overlaps of Poincaré boxes. Let $ \mathrm{pr}_{\mathcal{A}} : \widetilde{\mathcal{A}}^m\to\mathcal{A}^m$ be the obvious map.

Figure 10 Intersecting pair of pinching disks $ \{D_0, D_1\}$.

The images of the pinching disks in $ \widetilde{\mathcal{A}}^m$ by the projection $ \pi_x$ to $ \widetilde{\mathcal{S}}^m$ is called the pinching locus in $ \widetilde{\mathcal{S}}^m$ and denoted by $ P^m$. We fill up all holes in $ \widetilde{\mathcal{S}}^m$ ($ m=0, 1$) and choose a point from each hole which we call a center. Let $ C^m$ ($ m=0, 1$) be the set of centers in $ \widetilde{\mathcal{S}}^m$. We define $ \mathcal{H}^m$ to be the legal hull of $ P^m\cup C^m$ in $ \widetilde{\mathcal{S}}^m$ (just as in item (v) of Sect. 1) and then replace every point $ c\in \mathcal{H}^m$ which belongs to $ C^m$ by a loop to obtain a tree $ \widetilde{\mathcal{T}}^m$ decorated with loops. The map $ \tilde{\sigma}$ naturally induces a map $ \tilde{\tau} : \widetilde{\mathcal{T}}^1\to\widetilde{\mathcal{T}}^0$ up to homotopy. One can also define a map $ \tilde{\iota}_{\mathcal{T}} : \widetilde{\mathcal{T}}^1\to\widetilde{\mathcal{T}}^0$ up to homotopy which is the identity on $ \widetilde{\mathcal{T}}^0$ and smashing each connected component of $ \widetilde{\mathcal{T}}^1{\setminus}\widetilde{\mathcal{T}}^0$ to a point. We say that two points $ t$ and $ t'$ in $ P^m$ form a pinching pair and denoted as $ t\approx_m t'$ if there exist a pair of pinching disks $ \{D, D'\}$ in $ \widetilde{\mathcal{A}}^m$ so that $ t=\pi_x(D)$ and $ t'=\pi_x(D')$.

Now we construct a topological model for the Julia set starting from a Hubbard tree ([Ishii2011], [Ishii2014]). Consider first the shift map on the orbit space: \begin{eqnarray*} \tilde{\tau}^{\pm\infty} : \widetilde{\mathcal{T}}^{\pm\infty}\longrightarrow\widetilde{\mathcal{T}}^{\pm\infty} \end{eqnarray*} of the Hubbard tree $ \tilde{\iota}_{\mathcal{T}}, \tilde{\tau} : \widetilde{\mathcal{T}}^1\to \widetilde{\mathcal{T}}^0$. For $ \underline{t}=(t_n)_{n\in\mathbb{Z}}$ and $ \underline{t}'=(t'_n)_{n\in\mathbb{Z}}$ in $ \widetilde{\mathcal{T}}^{\pm\infty}$, we define $ \underline{t}\approx_{\pm\infty}\underline{t}'$ if either $ t_i=t'_i$ or $ t_i\approx_1t'_i$ holds for any $ i\in\mathbb{Z}$. We also write $ \underline{t}\sim_{\pm\infty}\underline{t}'$ if there exist a sequence of points $ \underline{t}=\underline{t}^0, \underline{t}^1, \dots, \underline{t}^k=\underline{t}'$ in $ \widetilde{\mathcal{T}}^{\pm\infty}$ with $ \underline{t}^j\approx_{\pm\infty}\underline{t}^{j+1}$ for all $ 0\leq j\leq k-1$. Then, $ \sim_{\pm\infty}$ defines an equivalence relation in $ \widetilde{\mathcal{T}}^{\pm\infty}$, hence the factor map: \begin{eqnarray*} \tilde{\tau}^{\pm\infty}/_{\sim_{\pm\infty}} : \widetilde{\mathcal{T}}^{\pm\infty}/_{\sim_{\pm\infty}}\longrightarrow \widetilde{\mathcal{T}}^{\pm\infty}/_{\sim_{\pm\infty}} \end{eqnarray*} of $ \tilde{\tau}^{\pm\infty}$ is well-defined. This is what we called the limit-quotient model in [Ishii2014].   × ⁹

The following theorem states that the dynamics of $ f$ is reconstructed from its Hubbard tree.

The proof of this result is based on the method of homotopy shadowing ([Ishii and Smillie2010]).

Here we present two examples of hyperbolic systems and their Hubbard trees. The first one called the Basilica-Horseshoe map denoted by $ f=f_{\mathrm{BH}}$ consists of three Poincaré boxes which are mapped with each other as described in Fig. 11. This looks similar to the non-planar cubic Hénon map in Fig. 9, but we merge the right-upper boxes $ \mathcal{A}_1$ and $ \mathcal{A}_2$ in Fig. 9 into one box (and denote it by $ \mathcal{A}_2$ in Fig. 11). The map $ f_{\mathrm{BH}}$ has a unique attractive cycle of period two and its Julia set is disconnected. Note that the restriction $ f_{\mathrm{BH}} : \mathcal{A}_2\cap f^{-1}_{\mathrm{BH}}(\mathcal{A}_1)\to\mathcal{A}_1$ looks like the Horseshoe map, and the restriction $ f_{\mathrm{BH}} : \mathcal{A}_3\cap f^{-1}_{\mathrm{BH}}(\mathcal{A}_1)\to\mathcal{A}_1$ looks like the Basilica map. Figure 12 describes the Hubbard tree for $ f_{\mathrm{BH}}$.

Figure 11 Poincaré boxes for the Basilica-Horseshoe map.

Figure 12 Hubbard tree for the Basilica-Horseshoe map $ f_{\mathrm{BH}}$ .

Figure 13 Poincaré boxes for the Airplane-Basilica map.

Figure 14 Hubbard tree for the Airplane-Basilica map $ f_{\mathrm{AB}}$ .

The second one called the Airplane-Basilica map denoted by $ f=f_{\mathrm{AB}}$ consists of three Poincaré boxes which are mapped with each other as described in Fig. 13. The map $ f_{\mathrm{AB}}$ has two attractive cycles of period two and three respectively, and its Julia set is connected. Note that the map $ f_{\mathrm{AB}}$ looks like a mixture of the Airplane map and the Basilica map. Figure 14 describes the Hubbard tree for $ f_{\mathrm{AB}}$.

In this section we introduce the notion of the iterated monodromy groups for a class of hyperbolic polynomial diffeomorphisms of $ \mathbb{C}^2$. Here we formulate an iterated monodromy group as an inverse semigroup action on certain quotient space of the disjoint union of several preimage trees (see Appendix of [Ishii2014] for the generality on inverse semigroup actions). In the original paper [Ishii2014] we defined this notion in terms of its Hubbard tree in an algorithmic way. Here, however, we give an intuitive definition starting from the data of a family of Poincaré boxes with overlaps. This definition is purely geometric and we explain it by the Basilica-Horseshoe map $ f_{\mathrm{BH}}$ appeared in the previous section.

Let $ A_x$ and $ A_y$ be connected open subsets of $ \mathbb{C}$ with $ A_y$ simply connected and let $ \mathcal{A}=A_x\times A_y$ be a Poincaré box. Let $ \iota_{\mathcal{A}}, f : \mathcal{A}\cap f^{-1}(\mathcal{A})\to \mathcal{A}$ be a hyperbolic system obtained as a small perturbation of an expanding polynomial map $ p : A_x\cap p^{-1}(A_x)\to A_x$ of degree $ d\geq 2$. By an appropriate change of coordinates (see Subsection 5.1 of [Ishii and Smillie2010]) we have $ p\circ \mathrm{pr}_x=\mathrm{pr}_x\circ f$, where $ \mathrm{pr}_x : \mathcal{A}\to A_x$ is the $ x$-projection. Then, for any vertical disk $ V$ in $ \mathcal{A}$, the slice $ V\cap f^n(\mathcal{A})$ has a nested structure so that $ V\cap J^0_-(\tilde{f})$ becomes a Cantor set, where \begin{eqnarray*} J_-^0(f)\equiv \bigcap_{n\geq 0}(f\circ \iota_{\mathcal{A}}^{-1})^n(\mathcal{A}). \end{eqnarray*}

Hence, the projection $ \mathrm{pr}_x : J^0_-(\tilde{f})\to A_x$ is a fibration with Cantor fibers, and the Cantor fiber over $ x\in A_x$ can be identified with the "ideal boundary" of the preimage tree of $ p$ rooted at $ x$. With this identification, the action of the iterated monodromy group of $ p$ on $ T$ is interpreted as the holonomy group action on a Cantor fiber. Moreover, we note that each connected component of $ V\cap f^n(\mathcal{A})$ can be identified with certain subtree of $ T$.

Next consider the disjoint model $ \tilde{\iota}_{\mathcal{A}}, \tilde{f} : \widetilde{\mathcal{A}}^1\to\widetilde{\mathcal{A}}^0$ of a hyperbolic system $ \iota_{\mathcal{A}},f : \mathcal{A}^1\to\mathcal{A}^0$ with several Poincaré boxes, where $ \widetilde{\mathcal{A}}^0\equiv\bigsqcup_{i\in\Sigma^0}\mathcal{A}_i$ and $ \mathcal{A}^0\equiv\bigcup_{i\in\Sigma^0}\mathcal{A}_i$ (see Sect. 5.2 for the definition of $ \widetilde{\mathcal{A}}^1$ and $ \mathcal{A}^1$). Then, the lamination in each box $ J^0_-(\tilde{f})\cap \mathcal{A}_i$ looks similar to the previous case, where \begin{eqnarray*} J_-^0(\tilde{f})\equiv \bigcap_{n\geq 0}(\tilde{f}\circ \tilde{\iota}_{\mathcal{A}}^{-1})^n(\widetilde{\mathcal{A}}^0) \end{eqnarray*} (and similar definition for $ J_-^1(\tilde{f})$). To the lamination in $ J^0_-(\tilde{f})\cap \mathcal{A}_i$ we can associate a preimage tree $ T_i$. Note that since the degree of the map varies from boxes to boxes, the preimage trees are no more regular but "SFT-like".

Finally, we define the iterated monodromy group for a hyperbolic system $ \iota_{\mathcal{A}}, f : \mathcal{A}^1\to \mathcal{A}^0$ with overlapping Poincaré boxes $ \mathcal{A}^0\equiv\bigcup_{i\in\Sigma^0}\mathcal{A}_i$ and $ \mathcal{A}^1=\bigcup_{(i, j)\in \Sigma^1}\mathcal{A}_i\cap f^{-1}(\mathcal{A}_j)$. To do this, we first need to define a space on which the iterated monodromy group acts. Note that in this case the laminations in $ \mathrm{pr}_{\mathcal{A}}(J^m_-(\tilde{f})\cap \mathcal{A}_i)$ are overlapping with each other. Since a geometric interpretation of an iterated monodromy group is the holonomy group action for the laminations, we need to understand how these laminations are continued from boxes to boxes. Each section of the lamination is identified with the ideal boundary of the corresponding preimage tree, so the overlapping data should be described by certain identification between the preimage trees.

Below we explain this identification rule for the Basilica-Horseshoe map. In this process, the following principle is important which extends the observation for the case of a single Poincaré box; there is a one-to-one correspondence between a subtree $ T_{i_N}$ in $ T_{i_0}$ of the form: \begin{eqnarray*} b_{i_0}\longleftarrow b'_{i_1}\longleftarrow \ \cdots \ \longleftarrow b^{(N-1)}_{i_{N-1}}\leftarrow T_{i_N} \end{eqnarray*} and a connected component of \begin{eqnarray*} V_{b_{i_0}}\cap \tilde{f}(\mathcal{A}_{i_1})\cap \cdots \cap \tilde{f}^{N-1}(\mathcal{A}_{i_{N-1}})\cap \tilde{f}^N(\mathcal{A}_{i_N}), \end{eqnarray*} where $ b_i$ is the base-point of a connected component $ \mathcal{T}_i$ of $ \mathcal{T}^0$ (hence it is the root of the preimage tree $ T_i$) and $ V_{b_i}=\mathrm{pr}_x^{-1}(b_i)$ is the vertical disk in $ \mathcal{A}_i$ over $ b_i$.

First note that the transitions between the Poincaré boxes for this map are described in Fig. 9 and hence its transition diagram is given in Fig. 15.

Step 1: $ T_2$ can be seen as a subtree of $ T_1$. The two subtrees $ T'_2$'s in the first level of $ T_1$ (the zeroth level is the root) correspond to the two connected components of $ V_{b_1}\cap \tilde{f}(\mathcal{A}_2)$ denoted by $ D'_{b_1}$ and $ E'_{b_1}$ respectively. Similarly, the subtree $ T'_2$ in the first level of $ T_2$ corresponds to $ D'_{b_2}=V_{b_2}\cap \tilde{f}(\mathcal{A}_2)$. Then, one of $ \mathrm{pr}_{\mathcal{A}}(D'_{b_1})$ and $ \mathrm{pr}_{\mathcal{A}}(E'_{b_1})$ coincides with $ \mathrm{pr}_{\mathcal{A}}(D'_{b_2})$ in $ \mathcal{A}$ by letting $ \mathrm{pr}_{\mathcal{A}}(V_{b_1})\cap\mathrm{pr}_{\mathcal{A}}(V_{b_2})\ne\emptyset$. Assume that $ D'_{b_1}$ is such a connected component and identify the corresponding subtree $ T'_2$ in the first level of $ T_1$ with the subtree $ T'_2$ in the first level of $ T_2$. Similarly we identify one of two $ T'_3$'s in the first level of $ T_1$ with $ T'_3$ in the first level of $ T_2$. This gives a bijective correspondence between the infinite paths ending at $ b_1$ in $ T_1$ through the union of $ T'_2$ and $ T'_3$ in the first level of $ T_1$ and the ones ending at $ b_2$ in $ T_2$. Hence $ T_2$ can be seen as a subtree of $ T_1$ (see Fig. 16).

Figure 15 Transition diagram for the Basilica-Horseshoe map.

Figure 16 Identifying the preimage trees for the Basilica-Horseshoe map .

The two subtrees $ T''_2$'s in the second level of $ T_3$ correspond to the two connected components of $ V_{b_3}\cap \tilde{f}(\mathcal{A}_1)\cap \tilde{f}^2(\mathcal{A}_2)$ denoted by $ D''_{b_3}$ and $ E''_{b_3}$ respectively. Similarly, the subtree $ T''_2$ in the second level of $ T_1$ corresponds to $ E''_{b_1}=V_{b_1}\cap \tilde{f}(\mathcal{A}_2)\cap \tilde{f}^2(\mathcal{A}_2)$. Then, one of $ \mathrm{pr}_{\mathcal{A}}(D''_{b_3})$ and $ \mathrm{pr}_{\mathcal{A}}(E''_{b_3})$ coincides with $ \mathrm{pr}_{\mathcal{A}}(E''_{b_1})$ by letting $ \mathrm{pr}_{\mathcal{A}}(V_{b_1})\cap\mathrm{pr}_{\mathcal{A}}(V_{b_3})\ne\emptyset$. Assume that $ D''_{b_3}$ is such a connected component and identify the corresponding subtree $ T''_2$ in the second level of $ T_3$ with the subtree $ T''_2$ in the second level of $ T_1$. Similarly we identify one of two $ T_3''$'s in the second level of $ T_3$ with $ T_3''$ in the second level of $ T_1$.

Now we repeat this procedure. Then, for most infinite paths in $ T_1$ ending at the root point one can find corresponding paths which are eventually identified by the procedure above. The only exception is the path: \begin{eqnarray*} b_1\longleftarrow b'_3\longleftarrow b''_1\longleftarrow b'''_3\longleftarrow \ \cdots \end{eqnarray*} in $ T_1$. We define its corresponding path in $ T_3$ to be the one: \begin{eqnarray*} b_3\longleftarrow b'_1\longleftarrow b''_3\longleftarrow b'''_1\longleftarrow \ \cdots. \end{eqnarray*} This gives a bijective correspondence between the infinite paths ending at $ b_1$ in $ T_1$ through the union of $ T'_2$ and $ T'_3$ in the first level of $ T_1$ and the ones ending at $ b_3$ in $ T_3$. Hence $ T_3$ can be seen as a subtree of $ T_1$ (see Fig. 16 again).

Step 3: Action of $ \mathrm{IMG}(\mathfrak{T})$ on $ T$. The discussion in the previous step defines an equivalence relation called the holonomy equivalence denoted by $ \sim_{\mathrm{holo}}$ in the disjoint union $ T_1\sqcup T_2\sqcup T_3$ of the preimage trees. Set \begin{eqnarray*} T\equiv \big(T_1\sqcup T_2\sqcup T_3\big)/_{\sim_{\mathrm{holo}}}. \end{eqnarray*} In the example discussed above, we identified $ T_2$ and $ T_3$ as subtrees of $ T_1$, therefore $ T=T_1$.

Let $ \mathrm{IMG}(\widetilde{\mathfrak{T}})_i$ be the iterated monodromy group associated to a connected component $ \mathcal{T}_i$ of $ \mathcal{T}^0$ acting on $ T_i$. Let $ \kappa : T_1\sqcup T_2\sqcup T_3\to T$ be the natural projection. Then, the action of $ \mathrm{IMG}(\widetilde{\mathfrak{T}})_i$ on $ T_i$ descends to an inverse semigroup action on $ T$ which is denoted by $ \kappa_{\ast}(\mathrm{IMG}(\widetilde{\mathfrak{T}})_i)$. We define \begin{eqnarray*} \mathrm{IMG}(\mathfrak{T}) \equiv \kappa_{\ast}(\mathrm{IMG}(\widetilde{\mathfrak{T}})_1) \odot \kappa_{\ast}(\mathrm{IMG}(\widetilde{\mathfrak{T}})_2) \odot \kappa_{\ast}(\mathrm{IMG}(\widetilde{\mathfrak{T}})_3), \end{eqnarray*} where $ G_1\odot G_2$ denotes the free product of two inverse semigroup actions $ G_1$ and $ G_2$ (see Section 7 of [Ishii2014]), and call it the iterated monodromy group of the Hubbard tree $ \mathfrak{T}$.

Let $ \Pi$ be the set of arrows in the diagram of Fig. 15 and put \begin{eqnarray*} \Pi^{-\infty}\equiv \Pi^{-\infty}_1\sqcup\Pi^{-\infty}_2\sqcup\Pi^{-\infty}_3, \end{eqnarray*} where \begin{eqnarray*} \Pi^{-\infty}_i\equiv \big\{(\pi_n)_{n\leq -1}\in\Pi^{-\mathbb{N}} : h(\pi_{n-1})=t(\pi_n) \mbox{ for all } n\leq -1 \mbox{ and } h(\pi_{-1})=i \big\} \end{eqnarray*} is the space of symbol sequences corresponding to the infinite paths in $ T_i$ ending at its root point. Hence, $ \Pi^{-\infty}$ can be seen as the "ideal boundary" of $ T$. Note that every element in $ \Pi^{-\infty}$ corresponds to an infinite path in $ T$ ending at some root point which we call a rooted path .

Let $ \Pi^{\pm\infty}$ be the set of bi-infinite paths in the diagram of Fig. 15.

The action of the iterated monodromy group $ \mathrm{IMG}(\mathfrak{T})$ on $ T^{\ast}/_{{\sim}_{\mathrm{holo}}}$ induces an action on $ \Pi^{\ast}/_{{\sim}_{\mathrm{holo}}}$ through the bijection $ \widetilde{\Lambda} : \Pi^{\ast}\to T^{\ast}$ which extends to an action on $ \Pi^{-\infty}/_{{\sim}_{\mathrm{holo}}}$. We call it the standard action of $ \mathrm{IMG}(\mathfrak{T})$ on $ \Pi^{-\infty}/_{{\sim}_{\mathrm{holo}}}$. For $ e, e'\in \Pi^{\pm\infty}/_{{\sim}_{\mathrm{holo}}}$ we write $ e\approx_{\mathrm{asym}} e'$ if $ \underline{\pi}\sim_{\mathrm{asym}}\underline{\pi}$ in the extended sense of Definition 1.17 (see Subsection 9.4 of [Bartholdi et al.2003] for more precise definition) for some representatives $ \underline{\pi}\in e$ and $ \underline{\pi}'\in e'$. The equivalence relation generated by $ \approx_{\mathrm{asym}}$ is called the asymptotic equivalence in $ \Pi^{\pm\infty}/_{{\sim}_{\mathrm{holo}}}$ and is denoted again by $ \approx_{\mathrm{asym}}$. Similarly, for $ E, E'\in \Pi^{\pm\infty}/_{\sim_{\mathrm{asym}}}$ we write $ E\approx_{\mathrm{holo}} E'$ if $ \underline{\pi}\sim_{\mathrm{holo}} \underline{\pi}'$ in the sense of Definition 6.5 for some representatives $ \underline{\pi}\in E$ and $ \underline{\pi}'\in E'$. The equivalence relation generated by $ \approx_{\mathrm{holo}}$ is called the holonomy equivalence in $ \Pi^{\pm\infty}/_{{\sim}_{\mathrm{asym}}}$ and is denoted again by $ \approx_{\mathrm{holo}}$.

We denote by $ s : \mathcal{S}_{\mathrm{IMG}(\mathfrak{T})}\to \mathcal{S}_{\mathrm{IMG}(\mathfrak{T})}$ the factor of the shift map on $ \Pi^{\pm\infty}$.

The proof of this result is based on the method of homotopy shadowing ([Ishii and Smillie2010]).

In his thesis [Oliva1998] introduced an algorithm for drawing gradient lines which allows the gradient lines to be drawn all the way until they land at $ J_f$. Thus the computer can determine whether two gradient lines land at the same point. These pictures are colored in a way that shows the solenoidal coding, and conversely, given a point of the solenoid, the computer can plot the corresponding ray. This gave Oliva the data (pairs of identified solenoidal points) which were the basis for his automaton. Figures 4.16, 4.17 and 4.18 in [Oliva1998] give plots where many pairs of rays were tested to corroborate his automaton.

The purpose of this section is to construct automata associated with iterated monodromy group actions for certain hyperbolic polynomial diffeomorphisms of $ \mathbb{C}^2$ along with [Ishii2014] (see also Lemma 1 in [Fried1987] for a related result). The idea of the construction is as follows. First, the inverse semigroup actions of $ \mathrm{IMG}(\widetilde{\mathfrak{T}})_i$ give rise to the direct sum of the inverse semigroup actions of \begin{eqnarray*} \mathrm{IMG}(\widetilde{\mathfrak{T}})\equiv \coprod_{i\in\Sigma^0}\mathrm{IMG}(\widetilde{\mathfrak{T}})_i \end{eqnarray*} on the disjoint union of preimage trees $ \bigsqcup_{i\in\Sigma^0}T_i$ (see Appendix of [Ishii2014] for the definition of the direct sum of inverse semigroup actions). We next reformulate the notion of holonomy equivalence in terms of an inverse semigroup action called the holonomy pinching group $ \mathcal{I}_{\mathfrak{T}}$. Intuitively the action of $ \mathrm{IMG}(\widetilde{\mathfrak{T}})$ describes the dynamics in the expanding direction and the action of $ \mathcal{I}_{\mathfrak{T}}$ describes the dynamics in the contracting direction. It is shown that the two actions are both self-similar and hence can define corresponding automata. We define the automaton for a polynomial diffeomorphisms of $ \mathbb{C}^2$ as certain power of the product of the two automata (Definition 6.12) and show that the quotient space with respect to the equivalence relation generated by the automaton is identical to $ \mathcal{S}_{\mathrm{IMG}(\mathfrak{T})}$ (Theorem 6.13).

To accomplish this procedure we first reformulate the notion of an automaton as follows and introduce the notion of the product of two automata. Let $ \Pi$ be a set called an alphabet .

It is often convenient to write an automaton $ \mathfrak{A}=(Q, q, \pi)$ as a pair of partially defined maps $ \tau=(q, \pi) : \Pi\times Q \dashrightarrow Q\times \Pi$. Given two automata $ \mathfrak{A}=(Q_{\mathfrak{A}}, q_{\mathfrak{A}}, \pi_{\mathfrak{A}})$ and $ \mathfrak{B}=(Q_{\mathfrak{B}}, q_{\mathfrak{B}}, \pi_{\mathfrak{B}})$ over a same alphabet $ \Pi$ we define their product $ \mathfrak{A}\mathfrak{B}=(Q_{\mathfrak{A}}\times Q_{\mathfrak{B}}, q_{\mathfrak{A}\mathfrak{B}}, \pi_{\mathfrak{A}\mathfrak{B}})$ over $ \Pi$, where $ \tau_{\mathfrak{A}\mathfrak{B}}=(q_{\mathfrak{A}\mathfrak{B}}, \pi_{\mathfrak{A}\mathfrak{B}}) : \Pi\times Q_{\mathfrak{A}}\times Q_{\mathfrak{B}}\dashrightarrow Q_{\mathfrak{A}}\times Q_{\mathfrak{B}}\times \Pi$ is given by the successive compositions: \begin{eqnarray*} \Pi\times Q_{\mathfrak{A}}\times Q_{\mathfrak{B}}\dashrightarrow Q_{\mathfrak{A}}\times\Pi\times Q_{\mathfrak{B}}\dashrightarrow Q_{\mathfrak{A}}\times Q_{\mathfrak{B}}\times\Pi \end{eqnarray*} of first $ \tau_{\mathfrak{A}}=(q_{\mathfrak{A}}, \pi_{\mathfrak{A}})$ and next $ \tau_{\mathfrak{B}}=(q_{\mathfrak{B}}, \pi_{\mathfrak{B}})$.

An automaton $ \mathfrak{A}=(Q, q, \pi)$ over $ \Pi$ defines a binary relation $ R_{\mathfrak{A}}\subset \Pi^{\mathbb{Z}}\times\Pi^{\mathbb{Z}}$. Let $ \underline{\pi}=(\pi_n)_{n\in\mathbb{Z}}, \underline{\pi}'=(\pi_n')_{n\in\mathbb{Z}}\in \Pi^{\mathbb{Z}}$ be two bi-infinite sequences. We say that $ (\underline{\pi}, \underline{\pi}')\in \Pi^{\mathbb{Z}}\times\Pi^{\mathbb{Z}}$ belongs to $ R_{\mathfrak{A}}$ if there exists a sequence $ (q_n)_{n\in\mathbb{Z}}\in Q^{\mathbb{Z}}$ so that $ q_{n+1}=q(\pi_n, q_n)$ and $ \pi'_n=\pi(\pi_n, q_n)$ hold.

From an automaton $ \mathfrak{A}=(Q, q, \pi)$ over an alphabet $ \Pi$, one can construct a doubly labeled directed graph called the Moore diagram of $ \mathfrak{A}$. Its vertex set is given by $ Q$, and for $ (\pi', q')\in \Pi\times Q$ in the common domain of $ \pi$ and $ q$ we draw an arrow from $ q(\pi', q')$ to $ q'$ labeled by the pair $ \pi' | \pi(\pi', q')$. Note that the direction of arrows defined here is opposite to the one in [Bartholdi et al.2003].

This definition is consistent with the binary relation $ R_{\mathfrak{A}}$. Namely, for $ \underline{\pi}, \underline{\pi}'\in\Pi^{\mathbb{Z}}$ we have $ (\underline{\pi}, \underline{\pi}')\in R_{\mathfrak{A}}$ if and only if $ \underline{\pi} \sim_{\mathfrak{A}} \underline{\pi}'$. When the binary relation $ R_{\mathfrak{A}}$ is an equivalence relation, we say that $ \mathfrak{A}$ generates the equivalence relation $ \sim_{\mathfrak{A}}$.

Given two binary relations $ R_1, R_2\subset \Pi^{\mathbb{Z}}\times\Pi^{\mathbb{Z}}$ in $ \Pi^{\mathbb{Z}}$, their product $ R_1R_2\subset\Pi^{\mathbb{Z}}\times\Pi^{\mathbb{Z}}$ is defined as follows; we say $ (\underline{\pi}^1, \underline{\pi}^2)\in R_1R_2$ iff there exists $ \underline{\delta}\in \Pi^{\mathbb{Z}}$ so that $ (\underline{\pi}^1, \underline{\delta})\in R_1$ and $ (\underline{\delta}, \underline{\pi}^2)\in R_2$ hold. Similarly, given two equivalence relations $ \sim_1, \sim_2\, \subset \Pi^{\mathbb{Z}}\times\Pi^{\mathbb{Z}}$ in $ \Pi^{\mathbb{Z}}$, their product $ \sim_1\sim_2$ is defined as the transitive closure of $ \sim_1$ and $ \sim_2$. In particular, we have \begin{eqnarray*} \sim_\mathfrak{A}\sim_\mathfrak{B}\ =\bigcup_{m> 0}(R_\mathfrak{A}R_\mathfrak{A})^m \end{eqnarray*} if two automata $ \mathfrak{A}$ and $ \mathfrak{B}$ generate the equivalence relations $ \sim_\mathfrak{A}$ and $ \sim_\mathfrak{B}$ respectively.

A key property in the construction of finite automata for Hénon maps is

Next we reformulate the holonomy equivalence in terms of certain inverse semigroup action. Choose $ i, i'\in\Sigma^0$ with $ i\ne i'$ so that $ E(i, i')\ne \emptyset$ holds, i.e. $ \mathcal{T}_i$ and $ \mathcal{T}_{i'}$ are identified at some points by $ \approx_{\mathfrak{L}^0}$. For $ \cdots\pi_{-2}\pi_{-1}\in\Pi^{-\infty}_i$ and $ \cdots\pi'_{-2}\pi'_{-1}\in\Pi^{-\infty}_{i'}$, we set $ \iota_{\{i, i'\}}(\cdots\pi_{-2}\pi_{-1})\equiv\cdots\pi'_{-2}\pi'_{-1}$ and $ \iota_{\{i, i'\}}(\cdots\pi'_{-2}\pi'_{-1})\equiv\cdots\pi_{-2}\pi_{-1}$ if either

1. (i) there exists $ N\leq -1$ so that the subpaths $ \cdots\pi_{N-1}\pi_{N}$ and $ \cdots\pi'_{N-1}\pi'_{N}$ of the paths $ \cdots\pi_{-2}\pi_{-1}$ and $ \cdots\pi'_{-2}\pi'_{-1}$ respectively are identified in $ T/_{\sim_{\mathrm{holo}}}$, or
2. (ii) $ \cdots\pi_{-2}\pi_{-1}$ and $ \cdots\pi'_{-2}\pi'_{-1}$ form an exceptional pair.

This gives an involution $ \iota_{\{i, i'\}}$ from a subset of $ \Pi^{-\infty}_i\sqcup\Pi^{-\infty}_{i'}$ to itself.

The holonomy pinching group $ \mathcal{I}_{\mathfrak{T}}$ acts on $ \Pi^{-\infty}$ faithfully and we denote its action of $ \iota\in\mathcal{I}_{\mathfrak{T}}$ as $ (\cdots\pi_{-2}\pi_{-1})^{\iota}$ for $ \cdots\pi_{-2}\pi_{-1}\in\Pi^{-\infty}$. Note that $ \mathcal{I}_{\mathfrak{T}}$ is similar to the holonomy pseudogroup in the foliation theory ([Candel and Conlon2000], [Nekrashevych2006]). The holonomy equivalence relation in Definition 6.5 can be then expressed in terms of $ \mathcal{I}_{\mathfrak{T}}$ as follows. Let $ \underline{\pi}=(\pi_n)_{n\in\mathbb{Z}}, \underline{\pi}'=(\pi_n')_{n\in\mathbb{Z}}\in\Pi^{\pm\infty}$. Then, $ \underline{\pi}\sim_{\mathrm{holo}}\underline{\pi}'$ holds iff for any $ n\in\mathbb{Z}$ there exists $ \iota_n\in\mathcal{I}_{\mathfrak{T}}$ so that $ (\cdots\pi_{n-1}\pi_n)^{\iota_n}=\cdots\pi_{n-1}'\pi_n'$ holds.

The notion of self-similarity of a group action is generalized to the setting of inverse semigroup actions in an appropriate way (see Definition 3.6 of [Bartholdi et al.2003]). One can show that the inverse semigroup actions of $ \mathcal{I}_{\mathfrak{T}}$ and $ \mathrm{IMG}(\widetilde{\mathfrak{T}})$ on $ \Pi^{-\infty}$ are both self-similar and contracting. Let $ \mathfrak{A}_{\mathrm{IMG}(\widetilde{\mathfrak{T}})}$ be the Moore diagram of the automaton $ (\mathcal{N}_{\mathrm{IMG}(\widetilde{\mathfrak{T}})}, \pi_{\mathrm{IMG}(\widetilde{\mathfrak{T}})}, q_{\mathrm{IMG}(\widetilde{\mathfrak{T}})})$. Also, we denote by $ \mathfrak{A}_{\mathcal{I}_{\mathfrak{T}}}$ the Moore diagram of the automaton $ (\mathcal{N}_{\mathcal{I}_{\mathfrak{T}}}, \pi_{\mathcal{I}_{\mathfrak{T}}}, q_{\mathcal{I}_{\mathfrak{T}}})$. Denote by $ \mathrm{card}(\Sigma^0)$ the cardinality of $ \Sigma^0$. A key observation is that the product $ \mathfrak{A}_{\mathrm{IMG}(\widetilde{\mathfrak{T}})}\mathfrak{A}_{\mathcal{I}_{\mathfrak{T}}}$ is $ \mathrm{card}(\Sigma^0)$-boundedly generating in $ \Pi^{\pm\infty}$ (Proposition 5.17 in [Ishii2014]). This motivates to define

Let $ \mathfrak{T}$ be a Hubbard tree and $ \Pi^{\pm\infty}$ be the subshift of finite type associated to it. Let $ \mathfrak{A}_{\mathfrak{T}}$ be the automaton for $ \mathfrak{T}$. Recall that for $ \underline{\pi}=(\pi_n)_{n\in\mathbb{Z}}, \underline{\pi}'=(\pi'_n)_{n\in\mathbb{Z}}\in \Pi^{\pm\infty}$, we write $ \underline{\pi}\sim_{\mathfrak{A}_{\mathfrak{T}}}\underline{\pi}'$ if there exists a bi-infinite path in $ \mathfrak{A}_{\mathfrak{T}}$ so that the sequence of labelings along the path is $ (\pi_n | \pi'_n)_{n\in\mathbb{Z}}$.

On the other hand, no tight automata theory for Hénon maps is established yet.

Write $ \Sigma_2\equiv \{A, B\}$ and denote by \begin{eqnarray*} \Sigma_2^{\mathbb{Z}}\equiv\big\{\cdots\varepsilon_{-1}\cdot\varepsilon_0\varepsilon_1\cdots : \varepsilon_i\in\Sigma_2\big\} \end{eqnarray*} the space of all two- sided symbol sequences over $ \Sigma_2$. We also consider the shift map $ \sigma : \Sigma_2^{\mathbb{Z}}\to\Sigma_2^{\mathbb{Z}}$ given by $ \sigma(\cdots\varepsilon_{-1}\cdot\varepsilon_0\varepsilon_1\cdots)\equiv \cdots\varepsilon_{-1}\varepsilon_0\cdot\varepsilon_1\cdots$.

We say that a complex Hénon map is a hyperbolic horseshoe on $ \mathbb{C}^2$ if its Julia set $ J_{c, b}$ is a hyperbolic set and $ f_{c, b} : J_{c, b}\to J_{c, b}$ is topologically conjugate to the shift map $ \sigma : \Sigma_2^{\mathbb{Z}}\to\Sigma_2^{\mathbb{Z}}$, where $ \Sigma_2^{\mathbb{Z}}$ is the space of bi-infinite symbol sequences with two symbols. The complex hyperbolic horseshoe locus is defined as \begin{eqnarray*} \mathcal{H}_{\mathbb{C}}\equiv\big\{(c, b)\in \mathbb{C}\times\mathbb{C}^{\times} : f_{c, b} \mbox{ is a hyperbolic horseshoe on } \mathbb{C}^2 \bigr\} \end{eqnarray*} Note that we do not know if $ \mathcal{H}_{\mathbb{C}}$ is connected. We define the shift locus $ \mathcal{S}_{c, b}$ for the complex Hénon family as the connected component of $ \mathcal{H}_{\mathbb{C}}$ containing the region $ \mathcal{H}_{\mathrm{OV}}\equiv\{(c, b)\in\mathbb{C}\times\mathbb{C}^{\times} : |c|> 2(1+|b|)^2\}$ found in [Oberste-Vorth1987] (see (1) of Corollary 5.7).

Fix $ (c_0, b_0)\in\mathcal{H}_{\mathrm{OV}}$. As in the case of polynomial maps in one complex variable (see item (viii) in Sect. 1), we have an anti-homomorphism: \begin{eqnarray*} \rho : \pi_1(\mathcal{S}_{c, b}, (c_0, b_0))\longrightarrow \mathrm{Aut}(\Sigma_2^{\mathbb{Z}}, \sigma) \end{eqnarray*} satisfying $ \rho(\gamma_1\cdot\gamma_2)=\rho(\gamma_2)\rho(\gamma_1)$, where $ \mathrm{Aut}(\Sigma_2^{\mathbb{Z}}, \sigma)$ is the group of homeomorphisms $ \tau : \Sigma_2^{\mathbb{Z}}\to \Sigma_2^{\mathbb{Z}}$ which commutes with the shift map $ \sigma$ on $ \Sigma_2^{\mathbb{Z}}$. We call $ \rho$ the monodromy presentation of the fundamental group $ \pi_1(\mathcal{S}_{c, b}, (c_0, b_0))$.

It is easy to see that the locus $ \mathcal{H}_{\mathbb{C}}$ is not simply connected; take a loop $ \gamma(t)=(c(t), b_0)\in\mathcal{H}_{\mathrm{OV}}$ where $ c(t)$ is a large loop surrounding the Mandelbrot set once with $ c(0)=c_0$, then $ \rho(\gamma)$ exchanges the two symbols $ A$ and $ B$. Moreover, [Arai2016] found an element $ \gamma\in\pi_1(\mathcal{S}_{c, b}, (c_0, b_0))$ so that $ \rho(\gamma)$ has infinite order.

One of the reasons why Conjecture 7.1 seems much more difficult to prove than Theorem 1.22 is that the group $ \mathrm{Aut}(\Sigma_2^{\mathbb{Z}}, \sigma)$ is "huge" compared to $ \mathrm{Aut}(\Sigma_2^{\mathbb{N}_0}, \sigma)$. For example, it is known that $ \mathrm{Aut}(\Sigma_2^{\mathbb{Z}}, \sigma)$ contains every finite group and the direct sum of countably many copies of $ \mathbb{Z}$, and no convenient system of generators is known.

Since the complex Hénon map is a diffeomorphism, it does not possess critical points in the usual sense. Therefore, one can not expect to obtain a "magic formula" as in item (ix) in Sect. 1 for the complex Hénon family. Here we propose two conjectures following the thesis of [Lipa2009] concerning the dynamics-parameter correspondence in the complex Hénon family. To do this, we investigate detailed combinatorial structure of the Mandelbrot set $ \mathcal{M}$ based on Theorem 1.26.

Let $ H$ be a hyperbolic component of the Mandelbrot set $ \mathcal{M}$. Theorem 1.26 implies that the union $ R_{\mathcal{M}}(\theta_{H}^-)\cup R_{\mathcal{M}}(\theta_{H}^+)\cup\{r_{\mathcal{M}}(H)\}$ divide the complex plane into two parts when $ H\ne\heartsuit$, and we also have $ R_{\mathcal{M}}(\theta_{\heartsuit}^-)\cup R_{\mathcal{M}}(\theta_{\heartsuit}^+)\cup\{r_{\mathcal{M}}(\heartsuit)\}=[1/4, +\infty)$.

One can associate the notion of a kneading sequence with each wake as follows [Lau and Schleicher1994], [Lipa2009] whose idea originates in Milnor–Thurston theory for maps of the interval ([Milnor and Thurston1977]). Let $ H$ be a hyperbolic component and let $ k(H)$ be the period of the unique attractive cycle of $ p_c$ with $ c\in H$. Given $ \theta\in\mathbb{T}$, we define $ K_{H}^+(\theta)=(i^+_n)_{n\geq 0}\in\Sigma_2^{\mathbb{N}_0}$ as \begin{eqnarray*} i^+_n\equiv \begin{cases} A & \mbox{if} \, \, 2^{n}\theta\in [\frac{\theta+1}{2}, \frac{\theta}{2}) \\ B & \mbox{if} \, \, 2^{n}\theta\in [\frac{\theta}{2}, \frac{\theta+1}{2}), \end{cases} \end{eqnarray*} for $ n\geq 0$, and $ K_{H}^-(\theta)=(i^-_n)_{n\geq 0}\in\Sigma_2^{\mathbb{N}_0}$ as \begin{eqnarray*} i^-_n\equiv \begin{cases} A & \mbox{if} \, \, 2^n\theta\in (\frac{\theta+1}{2}, \frac{\theta}{2}] \\ B & \mbox{if} \, \, 2^n\theta\in (\frac{\theta}{2}, \frac{\theta+1}{2}]. \end{cases} \end{eqnarray*} for $ n\geq 0$. One can then show that $ K^+(\theta_{H}^-)=K^-(\theta_{H}^+)$ holds if $ H\ne\heartsuit$.

We define the discarded kneading sequence of the wake $ \mathcal{W}_{H}$ as the first $ k(H)-1$ letters of the sequence $ K^+(\theta^-_{H})=K^-(\theta^+_{H})$ and denote it by $ \widehat{K}(\mathcal{W}_{H})$. We also set $ \widehat{K}(\mathcal{W}_{\mathcal{\heartsuit}})$ to be the empty word $ \epsilon$.

Here is a list of examples:

• When $ H=\heartsuit$ is the Main Cardioid, we have $ \theta^-_{\heartsuit}=0$, $ \theta^+_{\heartsuit}=1$ and $ k(\heartsuit)=1$. Since $ K(\mathcal{W}_{\heartsuit})=A$ holds, we see $ \widehat{K}(\mathcal{W}_{\heartsuit})=\epsilon$.
• When $ H$ is the Basilica component, we have $ \theta^-_{H}=1/3$, $ \theta^+_{H}=2/3$ and $ k(H)=2$. Since $ K(\mathcal{W}_{H})=BA$ holds, we see $ \widehat{K}(\mathcal{W}_{H})=B$.
• When $ H$ is the Rabbit component, we have $ \theta^-_{H}=1/7$, $ \theta^+_{H}=2/7$ and $ k(H)=3$. Since $ K(\mathcal{W}_{H})=BBA$ holds, we see $ \widehat{K}(\mathcal{W}_{H})=BB$.
• When $ H$ is the Airplane component, we have $ \theta^-_{H}=3/7$, $ \theta^+_{H}=4/7$ and $ k(H)=3$. Since $ K(\mathcal{W}_{H})=BAA$ holds, we see $ \widehat{K}(\mathcal{W}_{H})=BA$.

Another consequence of Theorem 1.26 is that the Mandelbrot set $ \mathcal{M}$ has a tree-like structure. More precisely, it yields that either $ \mathcal{W}_{H}\supset\mathcal{W}_{H'}$, $ \mathcal{W}_{H}\subset\mathcal{W}_{H'}$ or $ \mathcal{W}_{H}\cap\mathcal{W}_{H'}=\emptyset$ holds for two hyperbolic components $ H$ and $ H'$ of $ \mathcal{M}$.

We remark that a wake is always conspicuous to itself.

Through her numerical experiments with SaddleDrop, S. Koch ([Lipa2009], [Koch2012]) observed the "splitting phenomenon" of the Mandelbrot set $ \mathcal{M}$. Based on this phenomenon she defined (naively) the notion of herds as follows.

Suppose first that $ b=0$ and look at the $ c$-plane in the parameter space (see the left picture in Fig. 196); we then see the Mandelbrot set $ \mathcal{M}$ in the $ c$-plane. If we change $ b$ slightly, we still have a Mandelbrot-like set in the corresponding $ c$-plane to which we still have a well-defined notion of wakes.

Let $ \mathcal{W}_{H}$ be a wake of the Mandelbrot-like set with $ k(H)= 2$. As we perturb $ b$ more, the wake $ \mathcal{W}_{H}$ seems to split into two different pieces (see the middle picture in Fig. 196). One, called the $ A$- herd of $ \mathcal{W}_{H}$, which contains all the wakes in $ \mathcal{W}_{H}$ whose discarded kneading sequences end in $ A$, moves to the direction in the $ c$-plane that $ b$ is perturbed in. The other, called the $ B$- herd of $ \mathcal{W}_{H}$, which contains all the wakes in $ \mathcal{W}_{H}$ whose discarded kneading sequence end in $ B$, moves in the opposite direction (see the right picture in Fig. 196).

Figure 17 The $ c$-planes with $ b=0$ ( left ), $ b=0.015i$ ( middle ) and $ b=0.05i$ ( right ) ([Lipa2009]) .

Figure 18 A loop surrounding the $ B$-herd of Airplane wake with $ b=0.05i$ (( left ) which splits to the $ BB$-herd and the $ AB$-herd with $ b=0.2+0.3i$ ( middle ) ([Lipa2009]) .

Let $ \mathcal{W}_{H'}$ be a wake of the Mandelbrot-like set with $ k(H')= 3$ and $ \mathcal{W}_{H'}\subset \mathcal{W}_{H}$. As we perturb $ b$ even more, the $ A$- herd of $ \mathcal{W}_{H}$ inside $ \mathcal{W}_{H'}$ splits into two pieces. One, called the $ AA$- herd of $ \mathcal{W}_{H'}$, which contains all the wakes in $ \mathcal{W}_{H'}$ whose discarded kneading sequences end in $ AA$, moves a bit farther to the direction in the $ c$-plane that $ b$ is perturbed in than the other, called the $ BA$- herd of $ \mathcal{W}_{H'}$, which contains all the wakes in $ \mathcal{W}_{H'}$ whose discarded kneading sequence end in $ BA$. Similarly, the $ B$- herd of $ \mathcal{W}_{H}$ inside $ \mathcal{W}_{H'}$ (see the left picture in Fig. 197) splits into two pieces; the $ AB$- herd of $ \mathcal{W}_{H'}$ and the $ BB$- herd of $ \mathcal{W}_{H'}$ (see the right picture in Fig. 197).

After one more splitting, we have 8 herds associated with discarded kneading sequences in the following order: $ AAB$, $ BAB$, $ BBB$, $ ABB$, $ ABA$, $ BBA$, $ BAA$, $ AAA$ (note that the parity of the number of the letter $ B$ flips the lexicographical order). In this way we obtain the notion of the $ \underline{v}$- herd of a wake $ \mathcal{W}_{H}$ for a word $ \underline{v}$ over the alphabet $ \Sigma_2=\{A, B\}$. According to [Lipa2009], this splitting phenomenon has been observed numerically using SaddleDrop to a depth of 5.

Below the length of a word $ \underline{w}$ over the alphabet $ \{A, B, \ast\}$ is denoted by $ |\underline{w}|$. Given a word $ \underline{w}$ over $ \{A, B, \ast\}$ containing exactly one $ \ast$, we will define a continuous map $ \tau_{\underline{w}} : \Sigma_2^{\mathbb{Z}} \to \Sigma_2^{\mathbb{Z}}$ as follows. Take a sequence $ \underline{\varepsilon}=(\varepsilon_n)_{n\in\mathbb{Z}}\in\Sigma_2^{\mathbb{Z}}$. If there exist $ k\in\mathbb{Z}$ with $ \varepsilon_k\cdots\varepsilon_{k+|\underline{w}|-1}=\underline{w}$ except for the digit of $ \ast$ in $ \underline{w}$, we replace the letter in the corresponding digit in $ \underline{w}$ to the opposite one (i.e. $ A$ to $ B$ and $ B$ to $ A$). If there is no such $ k\in\mathbb{Z}$, $ \underline{\varepsilon}$ is left unchanged. By operating the above procedure to all possible $ k\in\mathbb{Z}$, we obtain a new sequence denoted by $ \tau_{\underline{w}}(\underline{\varepsilon})\in \Sigma_2^{\mathbb{Z}}$.

Now we are in position to state the first conjectures of Lipa (Conjecture 8.3 in [Lipa2009]).

When $ \mathcal{W}_{H}$ is the Airplane wake, we have $ K(\mathcal{W}_{H})=BAA$. He observed that the monodromy action $ \rho(\gamma)$ along the path $ \gamma$ in the left picture of Fig. 197 coincides with $ \tau_W$, where $ W=\{B\ast BAA\}$ (see Subsection 9.1 of [Lipa2009]). Similarly, he observed such coincidence of $ \rho(\gamma)$ for the two loops in the right picture of Fig. 197 and $ \tau_W$ with $ W=\{BB\ast BAA\}$ and $ W=\{AB\ast BAA\}$ respectively (see Subsection 9.2 of [Lipa2009]).

In Conjecture 8.4 of [Lipa2009], Lipa proposed the following "converse" to Conjecture 7.6.

For example, Lipa showed that $ \tau_W$ with $ W=\{A\ast BAA\}$ is not a compound marker automorphism and claims that he was not able to find a loop in the horseshoe locus that winds only around the $ A$-herd of the Airplane wake in Subsection 9.3 of [Lipa2009]. See Subsections 9.5 and 9.6 of [Lipa2009] for more examples and related discussions.

There are several programs to draw the Julia sets and parameter loci of the Henon family $ f_{a, b} : (x, y)\mapsto (x^2-a-by, x)$ in $ \mathbb{C}^2$ which are extremely useful for theoretical considerations of the dynamics. [FractalAsm2000] and [SaddleDrop2000] are created by Cornell Dynamics group (see Sect. 3.1). Ushiki ([Ushiki2012]) made programs called HénonExplorer and StereoViewer ; Figs. 11, 211, 214, 216 and 217 are drawn by these programs. These pictures present sets in four-dimensional space, and this software helps to visualize these pictures by showing them under different projections to the two-dimensional computer screen, or onto a stereo 3D viewer. An important feature is that the viewer allows the set to be viewed as it is rotated in 4D space. We recommend to visit his webpage ([Ushiki2012]) where more images can be found.

Figures 11 and 211 are drawn by HénonExplorer and Figs. 214, 216 and 217 are drawn by StereoViewer. Each of these figures shows a "cloud" of points which are colored whitish blue. These are the periodic points of periods up to $ 20$ for a Hénon map $ f$. Asymptotically, most periodic points are of saddle type, and the asymptotic distribution of the saddle periodic points gives the unique maximal entropy measure $ \mu_f$ of $ f$ (see [Bedford et al.1993b]). Thus this cloud of points approximates the set $ J_f^{\ast}$, which is defined as the support of the measure $ \mu_f$ (see Footnote 5 in Sect. 2.2). For the maps of Figs. 11 and 211, there are two saddle fixed points, and the surfaces shown in the figures are portions of the stable and unstable manifolds. Note that if we wish to display stable or unstable manifolds, we need to truncate them. This is because the stable and unstable manifolds of any saddle periodic point have homoclinic intersection, hence they return arbitrary close to the periodic point (see [Bedford et al.1993a]).

Figure 19 Close to a heteroclinic tangency ([Ushiki2012]).

In Fig. 11 there are two saddle fixed points close to each other which are bifurcated from a parabolic fixed point. The rectangular surface colored orange/brown is a piece of the stable manifold of one of the two saddle fixed points (there is a similar cyan-colored region, corresponding to the other saddle fixed point.) The darker shading corresponds to the values of $ G^-$; the darkest part indicates points of $ J^-$, where $ G^-(x, y)=0$. The other two surfaces are portions of the unstable manifolds, cut off so as not to obscure other portions of the picture. The shading corresponds to the value of $ G^+$, with the darkest part showing the points of $ J^+$. These saddle points in Fig. 11 are bifurcated from a map with a semi-parabolic fixed point. This bifurcation corresponds to a "parabolic implosion", which we can see because the fixed points show the spiraling behavior seen with the parabolic explosion of the familiar one-dimensional "cauliflower" Julia set. Such bifurcations were studied in detail by [Bedford et al.2012], and it is interesting to compare Fig. 11 with the figures in [Bedford et al.2012].

Figure 211 describes the dynamics of a Hénon map which has an almost tangential heteroclinic intersection. In the figure, the truncated surface colored green/yellow/red surrounding the one-dimensional-like filled Julia set in purple represents a part of the unstable manifold of a saddle fixed point. The other surface colored pink, which looks like an arch, represents the stable manifold of another saddle fixed point. As in Fig. 11, the darkness of the shading corresponds to the value of $ G^{\pm}$, and the darkest part indicates points of $ J^{\pm}$ where the surfaces intersect with the whitish blue cloud of points. The two surfaces intersect with each other almost tangentially at the "saddle point" of the arch, but this intersection does not belong to $ \mathbb{R}^2$.

Figure 20 Hénon map at the classical parameter ([Ushiki2012]).

Figure 214 describes the support $ J_f^{\ast}$ of the maximal entropy measure for the Hénon map $ f_{a, b}$ at the classical parameter $ (a, b)=(1.4, -0.3)$ ([Hénon1976]) seen from different directions. In each figure one can find the well-known strange attractor embedded in the picture which is the closure of the real unstable manifold of a saddle fixed point in the first quadrant of $ \mathbb{R}^2$. The attractor is decorated with portions of $ J_f^{\ast}$ not in the real plane. These are the "pruned branches" emanating into the imaginary directions. When we change the direction of our view-point as in the figures, these directions are twisted unexpectedly. These pruned branches become smaller when the parameter $ a$ increases, and eventually disappear when $ f_{a, b}$ becomes a horseshoe on $ \mathbb{R}^2$.

Figure 216 describes the dynamics of a real Hénon map which preserves the area in $ \mathbb{R}^2$. The large red curves are the KAM invariant circles of period two sitting in the real plane on which the twice iterate of $ f_{a, b}$ is topologically conjugate to an irrational rotation. One can also observe the nested structure of smaller KAM circles in red and the so-called chaotic sea in purple/yellow where the Lyapunov exponent is conjecturally strictly positive. Blue and green points represent orbits of randomly chosen initial points near the complex extensions of the KAM circles. They seem to present quasi-periodic motions, but we do not know if their orbit closures form tori.

Figure 21 Volume preserving Hénon map with KAM circles ([Ushiki2012]) .

Figure 22 Hénon map with coexisting attractive cycles ([Ushiki2012]) .

Figure 217 describes $ J_f^{\ast}$ of a Hénon map which possesses two attractive cycles of periods one and three respectively. The set $ J_f^{\ast}$ looks disconnected and hyperbolic (since it seems to satisfy the cigar condition ; see [Bedford and Smillie1999] for more details). The largest circle in $ J_f^{\ast}$ as well as its images belong to the boundary of the attractive basin of period one, and the other circles belong to the boundary of the attractive basins of period three. Locally $ J_f^{\ast}$ looks like the product of a portion of a one-dimensional-like Julia set and a Cantor set. However, since this map is quadratic and has two attractive cycles, one can show that it is non-planar (by assuming its hyperbolicity).

Let us explain some mathematical background behind Ushiki's programs. In these programs we first compute periodic points of the Hénon map $ f$ of period $ m$, say $ m\leq 20$. This means that we need to find all points $ (x, y)\in \mathbb{C}^2$ satisfying $ f^m(x, y)=(x, y)$. However, this is a polynomial equation of degree $ 2^m$ and, when $ m$ is large, it is practically impossible to find the zeros of the equation with such large degree by Newton's root-finding algorithm because the size of the attractive basins for the solutions can be extremely small.

To overcome this difficulty, Ushiki employed an algorithm of [Biham and Wenzel1989], [Biham and Wenzel1990], now called the BW-algorithm . For a given symbol sequence $ (\varepsilon_n)_{n\in\mathbb{Z}}\in \{+1, -1\}^{\mathbb{Z}}$, the algorithm consists of the following infinitely many ordinary differential equations:

As we saw above, the images drawn by HénonExplorer/StereoViewer are presented as objects in the two/three-dimensional space. However, the actual Julia sets are in $ \mathbb{C}^2$ which has real dimension four. As we observe especially through Fig. 214, it is hard to imagine how they are sitting in the four-dimensional space. Recently we have launched a 4D visualization project called Watch $ \_$ H This is a 4D visualization project launched by K. Anjyo, Z. Arai, H. Inou, Y. Ishii, S. Kaji and K. Tateiri.   × ¹⁰. The goal of the project is to express the images related to complex dynamics as fractal objects in the four-dimensional space and make an archive of such images (dynamical and parameter spaces for the Hénon family, dynamics on complex surfaces, etc). Towards this goal, we plan to employ a 3D virtual reality system, analyze the rotation in the four-dimensional space and develop new rendering techniques adapted to it.

Let $ f$ be a polynomial diffeomorphism of $ \mathbb{C}^2$ and let $ d\geq 2$ be its degree. We say that $ f$ is real if all the coefficients of $ f$ are real. In this case, the restriction $ f|_{\mathbb{R}^2} : \mathbb{R}^2\to\mathbb{R}^2$ is a well-defined dynamical system. It is known ([Friedland and Milnor1989]) that the topological entropy of $ f|_{\mathbb{R}^2}$ satisfies $ 0\leq h_{\mathrm{top}}(f|_{\mathbb{R}^2})\leq\log d$ for any real $ f$ of degree $ d$. We therefore say that a real polynomial diffeomorphism $ f$ attains the maximal entropy if $ h_{\mathrm{top}}(f|_{\mathbb{R}^2})=\log d$.

In [Bedford and Smillie2002] developed the theory of quasi-hyperbolicity . An important example of a quasi-hyperbolic map is a real polynomial diffeomorphism $ f$ with maximal entropy. Based on this theory, they have solved the so-called "first tangency problem" as follows.

Hereafter, we restrict our attention to the following form of the Hénon family: \begin{eqnarray*} f_{a, b} : (x, y)\longmapsto (x^2-a-by, x), \quad (a, b)\in\mathbb{R}\times\mathbb{R}^{\times} \end{eqnarray*} as dynamical systems on $ \mathbb{R}^2$. Let us call $ \mathbb{R}\times\mathbb{R}^{\times}$ the parameter space for the real Hénon family $ f_{a, b}$. When $ b\ne 0$ is fixed and $ a$ is large enough, $ f_{a, b}$ is a hyperbolic horseshoe on $ \mathbb{R}^2$, i.e. the restriction of $ f_{a, b}$ to its non-wandering set is uniformly hyperbolic and is topologically conjugate to the full shift with two symbols ([Devaney and Nitecki1979]). Such $ f_{a, b}$ attains the maximal entropy among the Hénon maps since we know $ 0\leq h_{\mathrm{top}}(f_{a, b})\leq \log 2$ for any $ (a, b)\in\mathbb{R}\times\mathbb{R}^{\times}$ by [Friedland and Milnor1989].

We are thus led to introduce the hyperbolic horseshoe locus : \begin{eqnarray*} \mathcal{H}_{\mathbb{R}}\equiv \bigl\{(a, b) \in\mathbb{R}\times\mathbb{R}^{\times} : f_{a, b} \mbox{ is a hyperbolic horseshoe on } \mathbb{R}^2 \bigr\} \end{eqnarray*} as well as the maximal entropy locus : \begin{eqnarray*} \mathcal{M}_{\mathbb{R}}\equiv \bigl\{(a, b) \in\mathbb{R}\times\mathbb{R}^{\times} : f_{a, b} \mbox{ attains the maximal entropy}\log 2 \bigr\}. \end{eqnarray*}

Note that $ \mathcal{H}_{\mathbb{R}}$ is open and $ \mathcal{M}_{\mathbb{R}}$ is closed in $ \mathbb{R}\times\mathbb{R}^{\times}$ (since $ h_{\mathrm{top}}(f_{a, b})$ is a continuous function of $ (a, b)$ by results of Katok and Newhouse; see page 110 of [Milnor1988]), hence $ \overline{\mathcal{H}_{\mathbb{R}}}\subset \mathcal{M}_{\mathbb{R}}$.

This result has been first obtained by [Bedford and Smillie2006] for the case $ |b|< 0.06$ and then generalized to all $ b\ne 0$ by Arai and the author ([Arai and Ishii2015]). We note that, when $ a=a_{\mathrm{tgc}}(b)$, the map $ f_{a, b}$ has exactly one orbit of either homoclinic ($ b> 0$) or heteroclinic ($ b< 0$) tangencies of stable and unstable manifolds of suitable saddle fixed points ([Bedford and Smillie2004]). The strategy of [Bedford and Smillie2006], [Arai and Ishii2015] is first to extend the dynamical and the parameter spaces over $ \mathbb{C}$, investigate their complex dynamical and complex analytic properties, and then reduce them to obtain conclusions over $ \mathbb{R}$. In the article ([Arai and Ishii2015]) we also employ interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences.

The statements described in Theorem 8.2 justify what were numerically observed at the beginning of 1980's by El Hamouly and Mira, Tresser, Ushiki and others. Figure 242 is obtained by joining two figures in the numerical work of [El Hamouly and Mira1981] and turning it upside down. There, the graph of the function $ a_{\mathrm{tgc}}$ is implicitly figured out by the right-most wedge-shaped curve.

As a consequence of Theorem 8.2, we obtain some global topological properties of the two loci. To state them, let us put $ \mathcal{H}^{\pm}_{\mathbb{R}}\equiv \mathcal{H}_{\mathbb{R}}\cap\{\pm b> 0\}$ and $ \mathcal{M}^{\pm}_{\mathbb{R}}\equiv \mathcal{M}_{\mathbb{R}}\cap\{\pm b> 0\}$. Below, we take the closure and the boundary of $ \mathcal{H}^{\pm}_{\mathbb{R}}$ and $ \mathcal{M}^{\pm}_{\mathbb{R}}$ in $ \{\pm b> 0\}$.

Figure 23 Bifurcation curves of the Hénon family [El Hamouly and Mira1981].

We note that this corollary can be regarded as a first step towards the understanding of an "ordered structure" in the parameter space for the Hénon family. Recall that in [Milnor and Tresser200] the monotonicity of the topological entropy for the cubic family (which has two parameters) is formulated as the connectivity of the isentropes. Therefore, the above result indicates a weak form of monotonicity of the entropy function $ (a, b)\mapsto h_{\mathrm{top}}(f_{a, b})$ at its maximal value.

It should be interesting to compare our results to the so-called anti-monotonicity theorem in [Kan et al.1992]. To be precise, we let $ h_t : \mathbb{R}^2\to\mathbb{R}^2$ ($ t\in\mathbb{R}$) be a one-parameter family of dissipative $ C^3$-diffeomorphisms of the plane and assume that $ h_{t_0}$ has a non-degenerate homoclinic tangency at certain parameter $ t=t_0$. Then, there are both infinitely many orbit-creation parameters and infinitely many orbit-annihilation parameters in any neighborhood of $ t_0\in\mathbb{R}$. It has been shown in [Bedford and Smillie2006] that for the one-parameter family of Hénon maps $ \{f_{a, b_{\ast}}\}_{a\in\mathbb{R}}$ with a fixed $ b_{\ast}> 0$ sufficiently close to zero, the map at $ a=a_{\mathrm{tgc}}(b_{\ast})$ taken from the boundary $ \partial\mathcal{H}^+_{\mathbb{R}}=\partial\mathcal{M}^+_{\mathbb{R}}$ has a non-degenerate homoclinic tangency. Of course, anti-monotonicity of some orbits does not necessarily imply anti-monotonicity of topological entropy. Nonetheless, the anti-monotonicity theorem suggests that, a priori, both $ \mathcal{H}_{\mathbb{R}}$ and $ \mathcal{M}_{\mathbb{R}}$ might have holes or several connected components separated from the largest one described in Corollary 8.3.

In their recent work [Bedford and Smillie2017] gave a characterization of the loci boundary for $ |b|< 0.06$ in terms of symbolic dynamics with respect to a family of three polydisks. A similar characterization of the loci boundary should be possible for all values of $ b$ in terms of symbolic dynamics with respect to a family of four polydisks for $ b> 0$ and a family of five polydisks for $ b< 0$ constructed in [Arai and Ishii2015].

Further problems and questions follow:

• Is the function $ a_{\mathrm{tgc}}$ monotone on $ \{b> 0\}$ and on $ \{b< 0\}$?
• Is the boundary of the zero-entropy locus piecewise real analytic (see [Gambaudo et al.1991])? This is true when $ |b|$ is close to zero ([Gambaudo et al.1989], [Carvalho et al.2005]).
• Is the complex hyperbolic horseshoe locus $ \mathcal{H}_{\mathbb{C}}$ for the complex Hénon family connected? We already know that it is not simply connected (see Sect. 7.1). Remark that since $ h_{\mathrm{top}}(f_{a, b})=\log 2$ for any $ (a, b)\in\mathbb{C}\times\mathbb{C}^{\times}$ by Theorem 2.2, the complex maximal entropy locus $ \mathcal{M}_{\mathbb{C}}$ is the entire parameter space $ \mathbb{C}\times\mathbb{C}^{\times}$.

One of the key steps in the proof of Theorem 8.2 was to show that the boundaries of the first tangency locus $ \partial\mathcal{H}^{\pm}_{\mathbb{R}}$ is surrounded by "tin cans" in the complexified parameter space (see Subsection 5.2 in [Arai and Ishii2015]). This condition has been verified by employing the interval Krawczyk method, an interval arithmetic version of Newton's root-finding algorithm. By modifying this argument together with the Schwarz Lemma in the parameter space we obtain the following estimate on the derivative of the function $ a_{\mathrm{tgc}}$.

Theorem 8.4 is applied to investigate ergodic properties of the real Hénon maps $ f_{a, b}$ at the first bifurcation parameters $ (a, b)\in\partial\mathcal{H}_{\mathbb{R}}^+$. Among others, we are interested in a variational characterization of equilibrium measures "at temperature zero". To state it, denote by $ M(f)$ the space of $ f$-invariant Borel probability measures of a Hénon map $ f$. An invariant measure $ \mu\in M(f)$ is called a $ (+)$- ground state if there exists an increasing sequence $ t_n\in\mathbb{R}$ with $ t_n\to+\infty$ as $ n\to\infty$ so that $ \mu$ is obtained as the weak limit of equilibrium measures for the potential function $ -t_n\log\|D_pf|E^u_p\|$, where $ E^u_p$ is the unstable direction of $ D_pf$ at $ p\in \mathbb{R}^2$. Let \begin{eqnarray*} \Lambda^u_{\mu}(f)\equiv \int\log \|D_zf|E^u_p\|d\mu(p), \end{eqnarray*} be the unstable Lyapunov exponent of $ f$ with respect to $ \mu\in M(f)$ and let \begin{eqnarray*} \Lambda^u(a, b)\equiv \inf_{\nu\in M(f_{a, b})}\Lambda^u_{\nu}(f_{a, b}). \end{eqnarray*}

An invariant measure $ \mu\in M(f)$ is called Lyapunov minimizing if it attains the infimum above. An invariant measure $ \mu\in M(f)$ is called entropy maximizing among the Lyapunov minimizing measures if it attains the supremum of the metric entropy $ h_{\nu}(f)$ over all Lyapunov minimizing measures $ \nu$ (see [Takahasi2016] for more detail).

Let $ U_{\delta}$ be the $ \delta$-neighborhood of the Chebyshev point $ (a, b)=(2, 0)$ in the parameter space of the real Hénon family $ f_{a, b}$. One can see that Theorem 8.4 yields a non-degeneracy condition in Theorem A (a) of [Takahasi2016] for the Hénon maps $ f_{a, b}$ with $ (a, b)\in\partial\mathcal{H}^+_{\mathbb{R}}\cap U_{\delta}$. As a consequence, we have the following variational characterization of the $ (+)$-ground states.

Corollary 8.5 indicates that a local geometric property of a complex parameter locus yields ergodic property of real Hénon maps.