The Renormalization Method and Quadratic-like Maps
نویسنده
چکیده
The renormalization of a quadratic-like map is studied. The threedimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the a priori complex bounds, the local connectivity of its Julia set is proved by using the three-dimensional Yoccoz puzzle. The generalized version of Sullivan’s sector theorem is discussed and is used to prove his result that the Feigenbaum quadratic polynomial has the a priori complex bounds and is unbranched. A dense subset on the boundary of the Mandelbrot set is constructed so that for every point of the subset, the corresponding quadratic polynomial is unbranched and has the a priori complex bounds.
منابع مشابه
Renormalization on One-dimensional Folding Maps
Some techniques and results in the renormalization theory of real and complex dynamical systems are summarized. The construction of the induced Markov map of [−1, 1] from a Feigenbaum-like map is presented. We show that this induced Markov map has bounded geometry. We discuss some property of infinitely renormalizable quadratic polynomials and show that the Julia set of an infinitely renormaliz...
متن کاملHyperbolicity of Renormalization of Critical Circle Maps
The renormalization theory of critical circle maps was developed in the late 1970’s–early 1980’s to explain the occurence of certain universality phenomena. These phenomena can be observed empirically in smooth families of circle homeomorphisms with one critical point, the so-called critical circle maps, and are analogous to Feigenbaum universality in the dynamics of unimodal maps. In the works...
متن کاملTime-Dependent Real-Space Renormalization Group Method
In this paper, using the tight-binding model, we extend the real-space renormalization group method to time-dependent Hamiltonians. We drive the time-dependent recursion relations for the renormalized tight-binding Hamiltonian by decimating selective sites of lattice iteratively. The formalism is then used for the calculation of the local density of electronic states for a one dimensional quant...
متن کاملCylinder renormalization for Siegel disks and a constructive Measurable Riemann Mapping Theorem
The boundary of the Siegel disk of a quadratic polynomial with an irrationally indifferent fixed point with the golden mean rotation number has been observed to be self-similar. The geometry of this self-similarity is universal for a large class of holomorphic maps. A renormalization explanation of this universality has been proposed in the literature. However, one of the ingredients of this ex...
متن کاملA Note on Quadratic Maps for Hilbert Space Operators
In this paper, we introduce the notion of sesquilinear map on Β(H) . Based on this notion, we define the quadratic map, which is the generalization of positive linear map. With the help of this concept, we prove several well-known equality and inequality...
متن کامل