On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies

Details On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies

2007-04-23

arxiv.org/abs/0704.2978v1

Mathematics

Dynamical Systems

17 pages, 9 figures. For supplemental materials, see http://www.math.kyoto-u.ac.jp/~arai/

Scientific paper

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex H\'enon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real H\'enon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex H\'enon map.

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