Monodromy And Henon Mappings

Abstract

We discuss the monodromy action of loops in the horseshoe locus of the Henon map on its Julia set. We will show that for a particular class of loops there is a certain combinatorially-defined subset of the Henon Julia set which must remain invariant under the monodromy action of loops in certain regions. We will then describe a conjecture for what the monodromy actions of these loops are as well as a possible connection between the algebraic structure of automorphisms of the full 2-shift and the existence of certain types of loops in the horseshoe locus.