The critical locus for complex Hénon maps (Q2851027)

Let \(f_a\) be a complex quadratic Hénon map, namely a biholomorphism of \(\mathbb{C}^2\) of the form NEWLINE\[NEWLINE f_a(x,y) = (x^2 + c -ay, x), NEWLINE\]NEWLINE with \(a\in \mathbb{C}\). Even if Hénon maps have no critical points in the usual sense, it is possible to define a dynamical analog of the critical points in this setting.NEWLINENEWLINEIn [\textit{J. H. Hubbard} and \textit{R. W. Oberste-Vorth}; Publ. Math., Inst. Hautes Étud. Sci. 79, 5--46 (1994; Zbl 0839.54029)] the authors introduced the rate of forward (resp. backward) escape function \(G_a^+\) (resp. \(G_a^-\)). This function is pluriharmonic on the set of points \(U_a^+\) [resp. \(U_a^-\)], whose forward (resp. backward) orbits tend to infinity. The level sets of \(G_a^+\) and \(G_a^-\) are foliated by Riemann surfaces. These natural foliations \(\mathcal{F}_a^+ \) and \(\mathcal{F}_a^-\) were introduced and extensively studied in [loc. cit.], and the critical locus \(C_a\) is defined as the set of tangencies between \(\mathcal{F}_a^+ \) and \(\mathcal{F}_a^-\).NEWLINENEWLINEIn the paper under review, the author gives an explicit topological model of the critical locus for Hénon maps such that \(x^2+ c\) has disconnected Julia set, and \(a\) is sufficiently small, justifying the picture conjectured by J. H. Hubbard. More precisely, under these hypotheses, the critical locus is a connected Riemann surface of infinite type, and it is homeomorphic to a union of countably many spheres, where the sphere \(S_i\) is attached to the sphere \(S_{i+j}\) by \(2^{j-1}\) handles, and the ``places where handles are attached'' converge to Cantor sets.