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Ronen Peretz

Publications and source records attributed to Ronen Peretz.

17 recordsLinked to original sources
arXiv2019

Extreme points and support points of conformal mappings

There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of $\Re L(\overline{z}_0(f(z))^2)$ up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by $f(z)$, for a support point $f(z)$ which maximizes $\Re L$. Here $L$ is a continuous linear functional on $H(U)$, the topological vector space of the holomorphic functions in the unit disk $U=\{z\in\mathbb{C}\,|\,|z|<1\}$. Such a support point is known to be a slit mapping and $f(z_0)$ is the tip of the slit $\mathbb{C}-f(U)$. The third demonstrates some properties of support points of the subspace $S_n$ of $S$. $S_n$ contains all the polynomials in $S$ of degree $n$ or less. For instance such a support point $p(z)$ has a zero of its derivative $p'(z)$ on $\partial U$.

math.CV↗
arXiv2017

On the automorphic group of an entire function

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory has already a long history that essentially started with two remarkable papers of Tatsujirô Shimizu that were published in 1931. The elements $ϕ(z)$ of the group are defined by the automorphic equation $f(ϕ(z))=f(z)$, were $f(z)$ is entire. Tatsujirô Shimizu also refers to the functions of this group as those functions that are determined by $f^{-1}\circ f$. He proved many remarkable properties of those automorphic functions. He indicated how they induce a beautiful geometric structure on the complex plane. Those structures were termed by Tatsujirô Shimizu, the system of normal polygonal domains, and the more refined system of the fundamental domains of $f(z)$. The last system if exists tiles up the complex plane with remarkable geometric tiles that are conformally mapped to one another by the automorphic functions. In the Ph.D thesis of the author, those tiles were also called the system of the maximal domains of $f(z)$. One can not avoid noticing the many similarities between this automorphic group and its accompanying geometric structures and analytic properties, and the more tame discrete groups that appear in the theory of hyperbolic geometry and also the arithmetic groups in number theory. This paper pursues further the theory initiated by Tatsujirô Shimizu, towards understanding global properties of the automorphic group, rather than just understanding the properties of the individual automorphic functions. We hope to be able in sequel papers to generalize arithmetic and analytic tools such as the Selberg trace formula, to this new setting.

math.CV↗
arXiv2017

A sharp version of Shimizu's theorem on entire automorphic functions

This paper develops further the theory of the automorphic group of non-constant entire functions. This theory essentially started with two remarkable papers of Tatsujirô Shimizu that were published in 1931. There are three results in this paper. The first result is that the ${\rm Aut}(f)$-orbit of any complex number has no finite accumulation point. The second result is an accurate computation of the derivative of an automorphic function of an entire function at any of its fixed points. The third result gives the precise form of an automorphic function that is uniform over an open subset of $\mathbb{C}$. This last result is a follow up of a remarkable theorem of Shimizu. It is a sharp form of his result. It leads to an algorithm of computing the entire automorphic functions of entire functions. The complexity is computed using an height estimate of a rational parameter discovered by Shimizu.

math.CV↗
arXiv2016

Composition arithmetic for locally one-to-one entire mappings

This paper describes a part of the factorization theory of the family of all the entire functions with non vanishing derivatives. In particular it proves that this family of mappings contains primes. This assures that this family of entire functions has two non degenerate fractal representations.

math.CV↗
arXiv2016

Applications of Steiner symmetrization to some extremal problems in geometric function theory

In this paper we investigate properties of the Steiner symmetrization in the complex plane. We use two recursive dynamic processes in order to derive some sharp inequalities on analytic functions in the unit disk. We answer a question that was asked by Albert Baernstein II, regarding the coefficients of circular symmetrization. We mostly deal with the Steiner symmetrization $G$ of an analytic function $f$ in the unit disk $U$. We pose few problems we can not solve. An intriguing one is that of the inequality $$ \int_{0}^{2π} |f(re^{iθ})|^{p}dθ\le\int_{0}^{2π} |G(re^{iθ})|^{p}dθ,\,\,0<p<\infty $$ which is true for $p=2$ (we prove) but can not be true for too large $p$. What is the largest such exponent or its supremum?

math.CV↗
arXiv2015

On an integral equation of Lieb

We prove that the weakly singular, non-linear convolution integral equation $\int_{\mathbb{R}^n}|x-y|^{-λ}f(y)dy=f(x)^{p-1}$, where $0<λ<n$, and $p=2n/(2n-λ)$ has at least two non-equivalent solutions. This answers a problem of Elliott Lieb. We also prove certain orthogonality relations among linear differential forms with constant coefficients related to the corresponding type of convolution operators. Finally, we discuss the regularity of the solutions of such non-linear integral equations over not necessarily bounded open subsets of $\mathbb{R}^n$.

math.CA↗
arXiv2014

Injectivity of the composition operators of étale mappings

We consider the semigroup of étale polynomial mappings $\mathbb{C}^2\rightarrow\mathbb{C}^2$ where the binary operation is composition. We prove that both the right and the left composition operators on this semigroup are injective. This is in contrast to the situation in the semigroup of the entire functions $\mathbb{C}\rightarrow\mathbb{C}$ which are locally injective, where the left composition operator is not injective. Our interest in the injectivity (of the left composition operator) results from a new approach to deal with the two dimensional Jacobian Conjecture. In this approach we construct a fractal like structure on the above (first) semigroup in order to use it to settle the conjecture (in preparation). Injectivity is crucial for that construction.

math.AG↗
arXiv2013

Picard theorems for Keller mappings in dimension two and the phantom curve

Let $F=(P,Q)\in\mathbb{C}[X,Y]^{2}$ be a polynomial mapping over the complex field $\mathbb{C}$. Suppose that $$ \det\,J_{F}(X,Y):=\frac{\partial P}{\partial X}\frac{\partial Q}{\partial Y}- \frac{\partial P}{\partial Y}\frac{\partial Q}{\partial X}=a\in\mathbb{C}^{\times}. $$ A mapping that satisfies the assumptions above is called a Keller mapping. In this paper we estimate the size of the co-image of $F$. We give a sufficient condition for surjectivity of Keller mappings in terms of its Phantom curve. This curve is closely related to the asymptotic variety of $F$.

math.AG↗
arXiv2012

On the structure of the semigroup of entire étale mappings

We hope to be able (in the future) to carefully analyze this structure and to tie the Jacobian Conjecture in dimension two to certain Zeta functions, thereby invoking a powerful arithmetic machinery to handle the two dimensional Jacobian Conjecture. Let us denote by ${\rm et}(\mathbb{C}^2)$ the semigroup of two dimensional Keller mappings. We would like to prove something like the following: a) That there exists an infinite index set $I$, and a family of mappings indexed by $I$, $\{F_i\,|\,i\in I\} \subset {\rm et}(\mathbb{C}^2)$ such that $$ {\rm et}(\mathbb{C}^2)={\rm Aut}(\mathbb{C}^2)\cup\bigcup_{i\in I} R_{F_i}({\rm et}(\mathbb{C}^2)), $$ where if $i\ne j$ then $R_{F_i}({\rm et}(\mathbb{C}^2))\cap R_{F_j}({\rm et}(\mathbb{C}^2))=\emptyset$. b) That the parallel representation to the representation described in (a) above holds true, this time with respect to the left composition operators $L_{G_j}$. These two claims will be the basis for a fractal structure on ${\rm et}(\mathbb{C}^2)$ because the pieces $R_{F_i}({\rm et}(\mathbb{C}^2))$ are similar to each other in the sense that they are homeomorphic, and we further have the scaling property of self-similarity, namely $R_F({\rm et}(\mathbb{C}^2))$ is homeomorphic to its proper subspace $R_{G\circ F}({\rm et}(\mathbb{C}^2))$ that is homeomorphic to its proper subspace $R_{H\circ G\circ F}({\rm et}(\mathbb{C}^2))$ etc . This is the right place to remark that the purpose of the current paper is to start and develop the parallel theory for entire functions in one complex variable. Results in this setting will hint that there are hopes to accomplish the above objective.

math.AG↗
arXiv2005

Iterated Images and the Plane Jacobian Conjecture

We show that the iterated images of a Jacobian pair stabilize; that is, the k-th iterates of a polynomial map of complex two-space to itself with a nonzero constant Jacobian determinant all have the same image for sufficiently large k. More generally, we obtain the same result for open polynomial maps of a closed algebraic subset X of complex N-space to itself that have finite coimage, and for cofinite subsets of such an X invariant under the map. We apply these results to obtain a new characterization of the two dimensional complex Jacobian conjecture related to questions of surjectivity.

math.AG↗
arXiv2005

Bounds on the Trace Mappings of LD-Fields

Bounds on the trace mappings defined on the Sobolev space W^{1,1}(Omega) and the space $LD(Omega)$ of integrable stains are obtained. Such bounds correspond to stress concentration--the ratio between the maximal stress in a body and the maximum of the traction applied to its boundary. The analysis leading to the bounds may be described in the mechanical context of stress theory and stress concentration.

math.AP↗
arXiv2002

A geometric inequality for circle packings

A geometric inequality among three triangles, originating in circle packing problems, is introduced. In order to prove it, we reduce the original formulation to the nonnegativity of a particular polynomial in four real indeterminates. Techniques based on sum of squares decompositions, semidefinite programming, and symmetry reduction are then applied to provide an easily verifiable nonnegativity certificate.

math.AG↗