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We study the dynamics of the renormalization operator acting on the space ofpairs (v,t), where v is a diffeomorphism and t belongs to [0,1], interpreted asunimodal maps x-->v(q_t(x)), where q_t(x)=-2t|x|^a+2t-1. We prove the so calledcomplex bounds for sufficiently renormalizable pairs with boundedcombinatorics. This allows us to show that if the critical exponent a is closeto an even number then the renormalization operator has a unique fixed point.Furthermore this fixed point is hyperbolic and its codimension one stablemanifold contains all infinitely renormalizable pairs.

Renormalization for critical orders close to 2N