No abstract

Consider the parameter space P λ ⊂ C 2 of complex Hénon maps H c,a (x, y) = (x 2 + c + ay, ax), a = 0 which have a semi-parabolic fixed point with one eigenvalue λ = e 2πip/q . We give a characterization of those Hénon maps from the curve P λ that are small perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in P λ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J + . The Julia set J + has a nice local description: inside a bidisk D r × D r it is a trivial fiber bundle over J p , the Julia set of the polynomial p, with fibers biholomorphic to D r . The Julia set J is homeomorphic to a quotiented solenoid.Theorem 1.1 (Structure Theorem). Let p(x) = x 2 + c 0 be a polynomial with a parabolic fixed point of multiplier λ = e 2πip/q . There exists δ > 0 such that for all parameters (c, a) ∈ P λ with 0 < |a| < δ there exists a homeomorphismThe map ψ depends on a, but we will show in Lemmas 12.7 and 12.8 that all maps ψ are conjugate to each other, for sufficiently small 0 < |a| < δ. Thus it does not matter which one we use and we can assume that the model map is ψ(ζ, z) = p(ζ), ζ − 2 z p (ζ) , for some > 0 independent of a. The function ψ is a solenoidal map in the sense of [HOV1]; it behaves like angle-doubling in the first coordinate, and contracts strongly in the second coordinate.Theorem 1.1 shows that J + ∩ V is a trivial fiber bundle over J p , the Julia set of the parabolic polynomial p(x) = x 2 + c 0 , with fibers biholomorphic to D r . The set J + is laminated by Riemann surfaces isomorphic to C. In fact, the current µ + supported on J + defined by Bedford and Smillie in [BS1] is laminar.

For any complex Hénon map H P, a : x y � → P(x) − ay x , the universal cover of the forward escaping set U + is biholomorphic to D × C, where D is the unit disk. The vertical foliation by copies of C descends to the escaping set itself and makes it a rather rigid object. In this note, we give evidence of this rigidity by showing that the analytic structure of the escaping set essentially characterizes the Hénon map, up to some ambiguity which increases with the degree of the polynomial P.

We use hyperbolicity of golden-mean renormalization of dissipative Hénon-like maps to prove that the boundaries of Siegel disks of sufficiently dissipative quadratic complex Hénon maps with golden-mean rotation number are topological circles. Conditionally on an appropriate renormalization hyperbolicity property, we derive the same result for Siegel disks of Hénon maps with all eventually periodic rotation numbers.