arXiv subjects
Publications and source records attributed to John Hubbard.
The "multicorn" is the connectedness locus of unicritical antiholomorphic polynomials $z\mapsto \bar r{z}^d+c$; the special case $d=2$ was named "tricorn" by Milnor. It appears as a natural local configuration in spaces of real cubic polynomials. We prove that no multicorn for $d\ge 2$ is pathwise connected, confirming a classical prediction based on numerical observations.
In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.
In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball such that the Hénon mapping extends continuously. That paper used a delicate analysis of some asymptotic expansions, for instance, to understand the structure of forward images of lines near infinity. The computations were quite difficult, and it is not clear how to generalize them to other rational maps. In this paper we will present an alternative approach, involving blow-ups rather than asymptotics. We apply it here only to Hénon mappings and their compositions, but the method should work quite generally, and help to understand the dynamics of rational maps $f:\Proj^2\ratto\Proj^2$ with points of indeterminacy. The application to compositions of Hénon maps proves a result suggested by Milnor, involving embeddings of solenoids in $S^3$ which are topologically different from those obtained from Hénon mappings.
Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The key invariant subsets are K_+/-, the sets of points with bounded forward images, J_+/- = the boundary of K_+/-, J = the union of J_+ and J_-, and K = the union of K_+ and K_-. In this paper we identify the topological structure of these sets when p is hyperbolic and |a| is sufficiently small, ie, when H is a small perturbation of the polynomial p. The description involves projective and inductive limits of objects defined in terms of p alone.