Renormalization for critical orders close to 2N

TL;DR

This paper investigates the renormalization operator for unimodal maps with critical exponents near even integers, establishing complex bounds and proving the existence and hyperbolicity of a unique fixed point for such maps.

Contribution

It introduces new complex bounds for renormalizable pairs with bounded combinatorics and proves the existence and hyperbolicity of a unique fixed point when the critical exponent is close to an even number.

Findings

Established complex bounds for renormalizable pairs.

Proved the existence of a unique fixed point near even critical exponents.

Showed the fixed point is hyperbolic with a codimension one stable manifold.

Abstract

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