Quasi-genericity of bifurcations to high dimensional invariant tori for maps

We consider a family of maps in a Banach spaceE near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle, the other part of the spectrum being strictly inside the unit disc. We focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantT ¹-circles into a family of invariantT ²-tori. We show that this problem needs a 3 dimensional parameter unfolding and that, for the complete maps, bifurcation occurs at points _,, where is the rotation number on the non-normally hyperbolicT ¹-circle, ande ^±2i are the eigenvalues of the constant matrix conjugated to the non-contracting part of the linearization on the normal fiber bundle overT ¹. Making some non-resonance and diophantine assumptions on (, ) leading to a positive measure Cantor set inT ², we show that in paraboloïdal regions of the 3 dim. parameter space we have clean bifurcations as for the truncated normal form. The complement of these regions forms a set of bubbles such as the ones obtained by Chenciner in Chen] for a codimension 2 problem for maps in ². The main tool here is a generalization for a matrix function onT ¹, close to a constant, of the quasi-conjugacy to a constant, modulo a minimum of additional parameters (moved quasi-conjugacy). For the infinite dimensional case we use aC decoupling result on the angular dependent linear parts into a contraction, still angular dependent, and another part quasi-conjugated to a constant matrix. This type of analysis applies for a wide range of problems, where truncated normal forms of the maps give bifurcations fromT ⁿ toT ⁿ⁺¹ tori, and this needs a (n+1)-dimensional parameter unfolding.We gratefully acknowledge the DRET (contrat 86/1445) who supported one of the authors (J.L.) during this work. This research has been also supported by the E.E.C. contract No. ST 2J-0316-C (EDB) on Mathematical problems in nonlinear Mechanics