Renormalisable Henon-like Maps and Unbounded Geometry

TL;DR

This paper proves that in a family of Henon-like maps, the set of parameters leading to Cantor sets with unbounded geometry is almost all parameters, highlighting typical complex structures in these dynamical systems.

Contribution

It demonstrates that for a broad class of Henon-like maps, unbounded geometric structures occur for almost all parameter values, extending understanding of their typical behavior.

Findings

Unbounded geometry occurs for full measure of parameters

Almost all maps in the family exhibit complex Cantor set structures

The result applies to strongly dissipative, infinitely renormalisable Henon-like maps

Abstract

We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable H\'enon-like maps, parametrised by a quantity called the `average Jacobian' $b$, the set of all parameters $b$ such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.