We prove the renormalization conjecture for circle diffeomorphisms with breaks, i.e. that the renormalizations of any two C 2+α -smooth circle diffeomorphisms with a break point, with the same irrational rotation number and the same size of the break, approach each other exponentially fast in the C 2 -topology. As a corollary, we obtain the following rigidity result: for almost all irrational rotation numbers, any two circle diffeomorphisms with a break, with the same rotation number and the same size of the break, are C 1 -smoothly conjugate to each other.

We construct a rigorous renormalization scheme for two degree-of-freedom analytic Hamiltonians, associated with Diophantine frequency vectors. We prove the existence of an attracting, integrable limit set of the renormalization. As an application of this renormalization scheme, we give a proof of a KAM theorem. We construct analytic invariant tori with Diophantine frequency vectors for near-integrable Hamiltonians in the domain of attraction of this limit set.

Link to this article: http://journals.cambridge.org/abstract_S0143385707000892 How to cite this article: HANS KOCH and SAŠA KOCIĆ (2008). Renormalization of vector elds and Diophantine invariant tori.Abstract. We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on T d × R . Each Diophantine vector ω ∈ R d determines an analytic manifold W of infinitely renormalizable vector fields, and each vector field on W is shown to have an elliptic invariant d-torus with frequencies ω 1 , ω 2 , . . . , ω d . Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting W to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

We study a directed polymer model defined on a hierarchical diamond lattice, where the lattice is constructed recursively through a recipe depending on a branching number b ∈ N and a segment number s ∈ N. When b ≤ s previous work [27] has established that the model exhibits strong disorder for all positive values of the inverse temperature β, and thus weak disorder reigns only for β = 0 (infinite temperature). Our focus is on the so-called intermediate disorder regime in which the inverse temperature β ≡ β n vanishes at an appropriate rate as the size n of the system grows. Our analysis requires separate treatment for the cases b < s and b = s. In the case b < s we prove that when the inverse temperature is taken to be of the form β n = β(b/s) n/2 for β > 0, the normalized partition function of the system converges weakly as n → ∞ to a distribution L( β) depending continuously on the parameter β. In the case b = s we find a critical point in the behavior of the model when the inverse temperature is scaled as β n = β/n; for an explicitly computable critical value κ b > 0 the variance of the normalized partition function converges to zero with large n when β ≤ κ b and grows without bound when β > κ b . Finally, we prove a central limit theorem for the normalized partition function when β ≤ κ b .