Postcritical sets and saddle basic sets for axiom A polynomial skew products on \(\mathbb {C}^2\) (Q2842230)

The paper under review deals with the relations between postcritical behaviors and saddle basic sets for Axiom A polynomial skew products on \(\mathbb C^2\). A polynomial skew product on \(\mathbb C^2\) is a map of the form \(f(z,w)=(p(z), q(z,w))\), where \(p(z)\) and \(q_z(w):=q(z,w)\) are polynomial of degree equal to or greater than \(2\). A map \(f\) satisfies Axiom A if its non-wandering set is hyperbolic, compact and the periodic points are dense in it. NEWLINENEWLINENEWLINEThe critical set \(C_{J_p}\) of \(f\) is the set \(\{(z,w)\in J_p\times \mathbb C: q_z'(w)=0\}\), where \(J_p\) is the Julia set of \(p\). For a subset \(X\subset \mathbb C^2\), let \(A(X)=\cap_{N\geq 0}\overline{\cup_{n\geq N} f^n(X)}\). Denote by \(A_{pt}(C_{J_p})=\overline{\cup_{x\in C_{J_p}} A(x)}\) the pointwise accumulation set of \(C_{J_p}\), and by \(A_{cc}(C_{J_p})=\overline{\cup_{C\in \text{Conn}(C_{J_p})} A(C)}\) the componentwise accumulation set of \(C_{J_p}\). Let \(\Lambda\) denote the closure of the set of saddle periodic points in \(J_p\times \mathbb C\). The saddle set \(\Lambda\) decomposes into a disjoint union of saddle basic sets \(\Lambda_i\), which are defined as compact invariant subsets of the non-wandering domain of \(f\) with dense orbit. Let \(C_i:=C_{J_p}\cap W^s(\Lambda_i)\), where \(W^s(\Lambda_i)\) is the stable set of \(\Lambda_i\), i.e., the set of points in \(\mathbb C^2\) which tend to \(\Lambda_i\) under iteration of \(f\).NEWLINENEWLINEThe author proves that \(A_{cc}(C_{J_p})=A_{pt}(C_{J_p})\) if and only if every connected component \(C\) of \(C_{J_p}\) is contained in the stable set of a saddle basic set \(C_i\). This improves previous results of L. DeMarco and S. Hruska. Then he proves that \(A(C_i)\) is closed if and only if \(A(C_i)=\Lambda_i\). As a consequence, \(A(C_{J_p})=A_{pt}(C_{J_p})\) if and only if \(C_i\) is closed for every \(i\). He also gives the characterization of the \(C_i\)'s which are open in terms of the unstable set of \(\Lambda_i\). Finally, he shows stability of the equalities \(A(C_{J_p})=A_{pt}(C_{J_p})\) and \(A_{cc}(C_{J_p})=A_{pt}(C_{J_p})\) in hyperbolic components.NEWLINENEWLINEThe paper contains several examples, among which, the author shows the existence of an Axiom A polynomial skew product of degree \(4\) such that \(J_p\) is neither connected nor totally disconnected and such that \(A_{cc}(C_{J_p})=A_{pt}(C_{J_p})\neq A(C_{J_p})\).