On Loops in the Hyperbolic Locus of the Complex Hénon Map and Their Monodromies
Abstract
Description
We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex Hénon 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 Hénon 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 Hénon map.
17 pages, 9 figures. For supplemental materials, see http://www.math.kyoto-u.ac.jp/~arai/
17 pages, 9 figures. For supplemental materials, see http://www.math.kyoto-u.ac.jp/~arai/