The Julia set of Hénon maps. (English) Zbl 1088.37019
Let \(H:\mathbb{C}^2\longrightarrow \mathbb{C}^2\) be a (complex) Hénon mapping of the form \(H=H_1\circ\cdots\circ H_n\), where \(n\geq 1\), \(H_i(z,w)=(P_i(z)+a_iw,b_iz)\) in which \(P_i\) is a polynomial of degree at least \(2\) and \(a_i\), \(b_i\) are nonzero complex constants. Associated to each Hénon mapping there is a natural invariant measure \(\mu\) with compact support \(J^*\). There is also a natural notion of the Julia set \(J\).
Main theorem in this paper is that if a complex Hénon mapping \(H\) is hyperbolic on \(J^*\), and if \(H\) is not volume preserving, then \(J=J^*\). It was proved by E. Bedford and J. Smillie [Invent. Math. 103, 69–99 (1991; Zbl 0721.58037)] that if \(H\) is uniformly hyperbolic when restricted to \(J\), then \(J=J^*\).
Main theorem in this paper is that if a complex Hénon mapping \(H\) is hyperbolic on \(J^*\), and if \(H\) is not volume preserving, then \(J=J^*\). It was proved by E. Bedford and J. Smillie [Invent. Math. 103, 69–99 (1991; Zbl 0721.58037)] that if \(H\) is uniformly hyperbolic when restricted to \(J\), then \(J=J^*\).
Reviewer: Pei-Chu Hu (Jinan)
MSC:
| 37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
| 32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |
| 37F50 | Small divisors, rotation domains and linearization in holomorphic dynamics |
Citations:
Zbl 0721.58037References:
| [1] | Bedford, E., Lyubich, M., Smillie, J.: Polynomial Diffeomorphisms of . IV. The measure of maximal entropy and laminar currents. Inv. Math. 112, 77–125 (1993) · Zbl 0792.58034 |
| [2] | Bedford, E., Smillie, J.: Polynomial Diffeomorphisms of : Currents. equilibrium measure and hyperbolicity. Inv. Math. 103, 69–99 (1991) · Zbl 0721.58037 |
| [3] | Bedford, E., Smillie, J.: Polynomial Diffeomorphisms of . II. Stable manifolds and recurrence. J. Amer. Math. Soc. 4, 657–679 (1991) · Zbl 0744.58068 |
| [4] | Bedford, E., Smillie, J.: Polynomial Diffeomorphisms of : VI. Connectivity of J. Ann. of Math. 148, 695–735 (1998) · Zbl 0916.58022 |
| [5] | Fornæss, J. E.: Real Methods in Complex Dynamics. in Real Methods in Complex and CR Geometry. D. Zaitsev, G. Zampieri (ed.) Springer-Verlag. Lecture Notes in Mathematics 1848, C.I.M.E Summer Course in Martina Franca. Italy 2002 · Zbl 1068.37028 |
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