Renormalization in the Hénon family. I: Universality but non-rigidity (Q2492844)

This extended and interesting paper deals with a class \({\mathcal H}\) of Hénon-like maps \(F\) in \(\mathbb{R}^2\) of the form \(F:(x,y)\mapsto(f(x)-\varepsilon(x,y),x)\), where \(f(x)\) is a unimodal map subject of certain regularity assumptions, and \(\varepsilon(x,y)\) is small. If \(f\) is renormalizable then the renormalization of \(F\) is defined as \(\mathbb RF=H^{-1} \circ(F^2|_U)\circ H\), where \(U\) is a certain neighborhood of the ``critical value'' \(v=(f(0),0)\) and \(H\) is an explicit nonlinear change of variables. The paper consists of 13 sections, which are headed followed here: 1. Introduction; 2. General notation and terminology; 3. Hénon renormalization. -- In this section, the class of Hénon-like maps is introduced and renormalization for such maps is defined. 4. Hyperbolicity of the Hénon renormalization operator. -- In this section, it is shown that the Hénon renormalization operator has a hyperbolic fixed-point \(F_*(x,y)=(f_*(x), x)\), where \(f_*\) is the fixed-point of the one-dimensional renormalization operator. 5. The critical Cantor set. -- In this section, the attracting set for infinitely renormalizable Hénon-like maps is studied. It is shown that its Hausdorff dimension is bounded from above by 0.73. 6. The average Jacobian. -- In this section, the average Jacobian \(b\) of an infinitely renormalizable Hénon-like map with respect to the unique invariant measure supported on its critical Cantor set is studied. 7. Universality around the tip. -- In this section (central in the paper), the shape of the renormalization \(\mathbb R^nF\) of the Hénon-like maps is derived. 8. Affine rescaling and quadratic change of variable. 9. Nonexistence of continuous invariant line fields. 10. Nonrigidity of the critical Cantor set. 11. Generic unbounded geometry. 12. Hölder geometry of the critical Cantor set. 13. Open problems. -- In this section, the six questions that naturally arise from the previous investigation are presented as open problems.