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Register a new userKAM-renormalization and Herman rings for 2D maps
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Authors
Michael Yampolsky
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In this note, we extend the renormalization horseshoe we have recently constructed with N. Goncharuk for analytic diffeomorphisms of the circle to their small two-dimensional perturbations. As one consequence, Herman rings with rotation numbers of bounded type survive on a codimension one set of parameters under small two-dimensional perturbations.
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We investigated several global behaviors of the weak KAM solutions $u_c(x,t)$ parametrized by $cin H^1(mathbb T,mathbb R)$. For the suspended Hamiltonian $H(x,p,t)$ of the exact symplectic twist map, we could find a family of weak KAM solutions $u_c(x,t)$ parametrized by $c(sigma)in H^1(mathbb T,mathbb R)$ with $c(sigma)$ continuous and monotonic and [ partial_tu_c+H(x,partial_x u_c+c,t)=alpha(c),quad text{a.e. } (x,t)inmathbb T^2, ] such that sequence of weak KAM solutions ${u_c}_{cin H^1(mathbb T,mathbb R)}$ is $1/2-$Holder continuity of parameter $sigmain mathbb{R}$. Moreover, for each generalized characteristic (no matter regular or singular) solving [ left{ begin{aligned} &dot{x}(s)in text{co} Big[partial_pHBig(x(s),c+D^+u_cbig(x(s),s+tbig),s+tBig)Big], & &x(0)=x_0,quad (x_0,t)inmathbb T^2,& end{aligned} right. ] we evaluate it by a uniquely identified rotational number $omega(c)in H_1(mathbb T,mathbb R)$. This property leads to a certain topological obstruction in the phase space and causes local transitive phenomenon of trajectories. Besides, we discussed this applies to high-dimensional cases.
In the nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophantine critical point of a hamiltonian function. I show that KAM versal deformation theory solves positively this conjecture.
In this paper, we consider the renormalization operator $mathcal R$ for multimodal maps. We prove the renormalization operator $mathcal R$ is a self-homeomorphism on any totally $mathcal R$-invariant set. As a corollary, we prove the existence of the full renormalization horseshoe for multimodal maps.
In the Nineties, Michel Herman conjectured the existence of a positive measure set of invariant tori at an elliptic diophatine critical point of a hamiltonian function. I construct a formalism for the UV-cutoff and prove a generalised KAM theorem which solves positively the Herman conjecture.
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
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