Related papers: Renormalization of $C^r$ H\'enon map : Two dimensi…
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the…
We demonstrate the existence of a global attractor A with a Cantor set structure for the renormalization of critical circle mappings. The set A is invariant under a generalized renormalization transformation, whose action on A is conjugate…
Renormalizations can be considered as building blocks of complex dynamical systems. This phenomenon has been widely studied for iterations of polynomials of one complex variable. Concerning non-polynomial hyperbolic rational maps, a recent…
This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable H\'enon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem…
According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to the…
The critical behavior for intermittency is studied in two coupled one-dimensional (1D) maps. We find two fixed maps of an approximate renormalization operator in the space of coupled maps. Each fixed map has a common relavant eigenvaule…
In this paper, we address a classical case of the Calder\'on (or conductivity) inverse problem in dimension two. We aim to recover the location and the shape of a single cavity $\omega$ (with boundary $\gamma$) contained in a domain…
We study $C^{d,r}$-H\'enon-like families $(f_{a\, b})_{a\, b}$ with two parameters $(a,b)\in \mathbb R^2$. We show the existence of an open set of parameters $(a,b)\in \mathcal D$, so that a renormalization chart conjugates an iterate of…
In this paper we show that the invariant Cantor set of period doubling type of any infinitely renormalizable area-preserving map in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in a…
If $Y$ is a properly embedded minimal surface in a convex cocompact hyperbolic 3-manifold $M$ with boundary at infinity an embedded curve $\gamma$, then Graham and Witten showed how to define a renormalized area $\calA$ of $Y$ via Hadamard…
Lorenz maps are maps of the unit interval with one critical point of order rho>1, and a discontinuity at that point. They appear as return maps of leafs of sections of the geometric Lorenz flow. We construct real a priori bounds for…
We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll}…
We prove that in the space of $C^r$ maps $(r=2,\ldots,\infty,\omega)$ of a smooth manifold of dimension at least 4 there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result…
It has been observed that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $\field{R}^2$. A renormalization approach has been used in a computer-assisted proof of existence of…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…
In this paper we generalize renormalization theory for analytic critical circle maps with a cubic critical point to the case of maps with an arbitrary odd critical exponent by proving a quasiconformal rigidity statement for renormalizations…
Correspondences between k-tuples of points are key in multiple view geometry and motion analysis. Regular transformations are posed by homographies between two projective planes that serves as structural models for images. Such…
We showed in part I (hep-th/9912092) that the Hopf algebra ${\cal H}$ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group $G$ and that the renormalized theory is obtained from the…
We continue our study [Ou4] of f-biharmonic maps and f-biharmonic submanifolds by exploring the applications of f-biharmonic maps and the relationships among biharmonicity, f-biharmonicity and conformality of maps between Riemannian…
It has been shown in (Gaidashev et al, 2010) and (Gaidashev et al, 2011) that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these…