Renormalization of three dimensional H??non map

Abstract

The three dimensional H??non-like map is defined on the cubic box domain. The geometric properties of Cantor attractor of F are studied. The Jacobian determinant of nth renormalized map is expressed asymptotically using the universal map. The set of the model maps, M is invariant class under renormalization. If there exist C^r invariant surfaces under model maps, then the geometric properties of Cantor attractor of F in M is involved with the same properties for the two dimensional H??non-like map. In particular, the non rigidity and the typical unbounded geometry of Cantor attractor are proved. Another invariant class under renormalization, N is defined by the particular equation of partial derivatives of third coordinate map of F. In contrast with the maps in M, the result of two dimensional H??non renormalization is not applied to the map in N. Instead the non linear scaling maps are analyzed in a direct way with recursive formulas. However, same geometric properties of Cantor attractor, in particular, non rigidity and typical unbounded geometry are also proved for the map in the class N.