Parameter exclusions in Henon-like systems

Stefano Luzzatto, Marcelo Viana

Published 2003-04-14Version 1

This survey is a presentation of the arguments in the proof that Henon-like maps f_a(x,y)=(1-a x^2,0) + R(a,x,y) with |R(a,x,y)|< b have a "strange attractor", with positive Lebesgue probability in the parameter "a", if the perturbation size "b" is small enough. We first sketch a "geometric model" of the strange attractor in this context, emphasising some of its key geometrical properties, and then focus on the construction and estimates required to show that this geometric model does indeed occur for many parameter values. Our ambitious aim is to provide an exposition at one and the same time intuitive, synthetic, and rigorous. We think of this text as an introduction and study guide to the original papers in which the results were first proved. We shall concentrate on describing in detail the overall structure of the argument and the way it breaks down into its (numerous) constituent sub-arguments, while referring the reader to the original sources for detailed technical arguments.