Abstract
AbstractWe give a new proof of a theorem of Hubbard and Oberste-Vorth (Real and Complex Dynamical Systems, pp. 89-132, 1995) for Hénon maps that are perturbations of a hyperbolic polynomial and obtain the Julia set $J^{+}$ J + inside a polydisk as the image of the fixed point of a contracting operator. We also give different characterizations of the Julia sets J and $J^{+}$ J + which prove useful for later applications.
Highlights
1 Introduction Fixed point theorems have found a lot of applications in dynamical systems in higher dimensions
In this article we give a description of the global structure of the Julia sets J and J+ of a dissipative hyperbolic Hénon map in C as the unique fixed point of a contracting operator in an appropriate function space
A complex Hénon map Hp,a : C → C is defined by Hp,a(x, y) = (p(x) + ay, ax), where p is a monic polynomial of degree d ≥
Summary
Fixed point theorems have found a lot of applications in dynamical systems in higher dimensions. In this article we give a description of the global structure of the Julia sets J and J+ of a dissipative hyperbolic Hénon map in C as the unique fixed point of a contracting operator in an appropriate function space. This provides an alternative proof of a well-known theorem of Hubbard and Oberste-Vorth [ ], which was one of the starting points (along with [ , ] and the works of Friedland and Milnor [ ], Bedford and Smillie [ – ], Fornæss and Sibony [ ], etc.) of more than two decades of research in dynamics in several complex variables. Hubbard and Oberste-Vorth [ ] studied the structure of the Julia sets J, J+, and J– for Hénon maps which are small perturbations of a hyperbolic polynomial p.
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