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 prove some new continuity results for the Julia sets J and J + of the complex Hénon map H c,a (x, y) = (x 2 + c + ay, ax), where a and c are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue (1 + t)λ, where λ is a root of unity and t is real and small in absolute value. These maps have a semi-parabolic fixed point when t is 0, and we use the techniques that we have developed in [RT] for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero |t|, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets J and J + depend continuously on the parameters as t → 0, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on J and J + when t is nonnegative.
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