Related papers: KAM-renormalization and Herman rings for 2D maps

It is well known that the mathematical structure underlying renormalization in perturbative quantum field theory is based on a Hopf algebra of Feynman diagrams. A precondition for this is locality of the field theory. Consequently, one…

For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here ``eigenvalue''…

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least…

We investigate the transfer of regularity between commutative, noetherian, local rings through a class of local homomorphisms which we call basically regular. We give numerical characterizations of these maps, investigate their behavior…

We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than $1$ but…

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to…

An algorithmic proof of the General Neron Desingularization theorem is given for $2$-dimensional local rings and morphisms with small singular locus.

A method of ``blocking'' triangulations that rests on the self-similarity feature of dynamically triangulated random manifolds is proposed. The method is used to define the renormalization group for random geometries. As an illustration,…

Perturbations due to round-off errors in computer modeling are discontinuous and therefore one cannot use results like KAM theory about smooth perturbations of twist maps. We elaborate a special approximation scheme to construct two smooth…

We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every…

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified…

We consider a renormalization transformation $R$ for skew-product maps of the type that arise in a spectral analysis of the Hofstadter Hamiltonian. Periodic orbits of $R$ determine universal constants analogous to the critical exponents in…

We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify…

The main purpose of this paper is to study restricted formal deformations of restricted Lie-Rinehart algebras in positive characteristic $p$. For $p>2$, we discuss the deformation theory and show that deformations are controlled by the…

A basic result in the theory of holomorphic functions of several complex variables is the following special case of the work of H. Cartan on the sheaf cohomology on Stein domains ([10], or see [14] or [16] for more modern treatments).

We prove an abstract KAM theorem adapted to space-multidimensional hamiltonian PDEs with regularizing nonlinearities. It applies in particular to the singular perturbation problem studied in the first part of this work.

In this article we prove that iterated renormalisations of $\mathcal{C}^r$ circle diffeomorphisms with $d$ breaks, $r>2$, with given size of breaks, converge to an invariant family of piecewise Moebius maps, of dimension $2d$. We prove that…

For supersymmetric gauge theories a consistent regularization scheme that preserves supersymmetry and gauge invariance is not known. In this article we tackle this problem for supersymmetric QED within the framework of algebraic…

We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…

Entanglement renormalization can be viewed as an encoding circuit for a family of approximate quantum error correcting codes. The logical information becomes progressively more well-protected against erasure errors at larger length scales.…