Related papers: Topology of the regular part for infinitely renorm…
We prove the a priori bounds for infinitely renormalizable quadratic polynomials for which we can find an infinite sequence of primitive renormalizations such that the ratios of the periods of successive renormalizations is bounded. This…
In this paper we give a combinatorial description of the renormlization limits of infinitely renormalizable unimodal maps with {\it essentially bounded} combinatorics admitting quadratic-like complex extensions. As an application we…
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…
We prove that non-hyperbolic non-renormalizable quadratic polynomials are expansion inducing. For renormalizable polynomials a counterpart of this statement is that in the case of unbounded combinatorics renormalized mappings become almost…
A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at…
We prove the uniform hyperbolicity of the near-parabolic renormalization operators acting on an infinite-dimensional space of holomorphic transformations. This implies the universality of the scaling laws, conjectured by physicists in the…
A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials…
In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the…
We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the…
The necessity of renormalization arises from the infinite integrals which are caused by the discrepancy between the orders of differential and integral operators in the four dimensional QFTs. Therefore in view of the fact that finiteness…
Systems of orthogonal polynomials whose recurrence coefficients tend to infinity are considered. A summability condition is imposed on the coefficients and the consequences for the measure of orthogonality are discussed. Also discussed are…
We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain…
We construct complex a-priori bounds for certain infinitely renormalizable Lorenz maps. As a corollary, we show that renormalization is a real-analytic operator on the corresponding space of Lorenz maps.
We study the boundary of the nonnegative trigonometric polynomials from the algebraic point of view. In particularly, we show that it is a subset of an irreducible algebraic hypersurface and established its explicit form in terms of…
We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.
In the paper we obtain the asymtotic number of integral quadratic polynomials with bounded heights and discriminants as the upper bound of heights tends to infinity.
In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.
We study the infimum of the renormalized volume for convex-cocompact hyperbolic manifolds, as well as describing how a sequence converging to such values behaves. In particular, we show that the renormalized volume is continuous under the…
The renormalization of a quadratic-like map is studied. The three-dimensional Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed. For an unbranched quadratic-like map having the {\sl a priori} complex bounds, the…
Spinor polynomials are polynomials with coefficients in the even sub-algebra of conformal geometric algebra whose norm polynomial is real. They describe rational conformal motions. Factorizations of spinor polynomial corresponds to the…