Dynamics of polynomial skew products on \(\mathbb{C}^2\)
Let \(f(z,w)=(p(z),q(z,w))\) be a polynomial skew product on \({\mathbb C}^2\), i.e, \(p\) and \(q\) are polynomials of degree \(d \geq 2\) such that \(p(z)=z^d+O(z^{d-1})\) and \(q(z,w)=w^d+O_z(w^{d-1})\). The \(n\)-th iterate of \(f\) is denoted by \(f^n\). This article is a detailed investigation to which extent the dynamical properties of polynomials on \(\mathbb C\) extend to such skew products. The first component \(p\) of \(f\) defines a dynamical system on \(\mathbb C\). The Green function \(G_p\) associated to \(p\) is defined by \(G_p(z) := \lim_{n\to\infty}{d^{-n}\log^+{|p^n(z)|}}\), and it measures the escape rate to infinity. The (compact) set \(K_p := \{z \in \mathbb C : G_p(z)=0\}\) consisting of all points with bounded orbit, is the filled Julia set of \(p\), and its boundary \(J_p\) is the Julia set of \(p\). In fact, \(G_p\) coincides with the Green function of \(K_p\) with pole at infinity. The harmonic measure \(\mu_p\) of \(K_p\) satisfies \(\text{supp}{\mu_p}=J_p\), it is invariant, ergodic, of maximal entropy, and it describes the distribution of periodic points of \(p\). The critical set of \(p\) is denoted by \(C_p\). In a similar way, there is a Green function \(G\) on \({\mathbb C}^2\) associated to \(f\) and an invariant ergodic measure \(\mu\) of maximal entropy. The author defines the support of \(\mu\) as the Julia set \(J_2\) of \(f\). The measure \(\mu\) coincides with the pluricomplex equilibrium measure on \(K\), the set of points in \({\mathbb C}^2\) with bounded orbit under \(f\). For \(z \in \mathbb C\), the restriction of \(f^n\) to a line \(\{z\} \times \mathbb C\) can be viewed as the composition of \(n\) monic polynomials on \(\mathbb C\). Therefore, we also have a critical set \(C_z\), a Green function \(G_z\), a probability measure \(\mu_z\), a filled Julia set \(K_z\) and a Julia set \(J_z\). There holds \(J_2 = \overline{\bigcup_{z \in J_p} \{z\} \times J_z}\), and the author deduces that \(J_2\) is the closure of the repelling periodic points of \(f\). A polynomial skew procuct \(f\) is called connected if \(J_p\) and \(J_z\) are connected sets for all \(z \in J_p\). The author shows that \(f\) is connected if and only if \(C_p \subset K_p\) and \(C_z \subset K_z\) for all \(z \in J_p\). Moreover, if \(f\) is connected, then \(J_2\) is connected, and so is \(J_z\) for all \(z \in \mathbb C\). The set of all polynomial skew products of degree \(d \geq 2\) may be identified with \({\mathbb C}^N\) for some positive integer \(N=N(d)\). If \(M_d\) denotes the subset of \({\mathbb C}^N\) corresponding to connected skew products, the author shows that \(M_d\) is compact. A further main object of this article is whether a polynomial skew product on \({\mathbb C}^2\) satisfies Axiom A (whose definiton is not recalled here). A polynomial \(p\) on \(\mathbb C\) satisfies Axiom A if and only if \(p\) is uniformly expanding on its Julia set \(J_p\), and this happens if and only if the closure of the postcritical set of \(p\) is disjoint from \(J_p\). The author shows that the latter equivalence statement does also hold for polynomial skew products, while a characterization of Axiom A in terms of expansion and in terms of the postcritical set is more complicated.
scientific article; zbMATH DE number 1337005