Showing papers on "Entire function published in 2020"

Wandering domains for entire functions of finite order in the Eremenko–Lyubich class

Abstract: Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.

Peacock patterns and resurgence in complex Chern-Simons theory

Abstract: The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite dimensional state-integral which is a holomorphic function of a complexified Planck's constant $\tau$ in the complex cut plane and an entire function of a complex parameter $u$. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear $q$-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte-Gaiotto-Gukov. We provide proofs of our statements regarding the $q$-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic $4_1$ and $5_2$ knots.

Inverse Sturm-Liouville problem with analytical functions in the boundary condition

Abstract: The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

Analytic structure on the spectrum of the algebra of symmetric analytic functions on $$L_\infty $$L∞

Abstract: We consider the subalgebra $$H_{bs}(L_\infty )$$ of analytic functions of bounded type on $$L_\infty [0,1]$$ which are symmetric, i.e. invariant, with respect to measurable bijections of [0, 1] that preserve the measure. Our main result is that $$H_{bs}(L_\infty )$$ is isomorphic to the algebra of all analytic functions on the strong dual of the space of entire functions on the complex plane $${\mathbb {C}}.$$ From this result we deduce that $$H_{bs}(L_\infty )$$ is a test algebra for Michael problem about the continuity of complex valued homomorphisms on Frechet algebras.

On the periodicity of transcendental entire functions

Abstract: According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\\geq 2$ is an integer.

An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition

Abstract: The Sturm-Liouville pencil is studied with arbitrary entire functions of the spectral parameter, contained in one of the boundary conditions. We solve the inverse problem, that consists in recovering the pencil coefficients from a part of the spectrum satisfying some conditions. Our main results are 1) uniqueness, 2) constructive solution, 3) local solvability and stability of the inverse problem. Our method is based on the reduction to the Sturm-Liouville problem without the spectral parameter in the boundary conditions. We use a special vector-functional Riesz-basis for that reduction.

A landing theorem for entire functions with bounded post-singular sets

Abstract: The Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Strong approximation of Gaussian $\beta$-ensemble characteristic polynomials: the edge regime and the stochastic Airy function

Abstract: We investigate the characteristic polynomials of the Gaussian $\beta$-ensemble for general $\beta>0$ through its transfer matrix recurrence. We show that the rescaled characteristic polynomial converges to a random entire function in a neighborhood of the edge of the limiting spectrum. This random entire function, called the stochastic Airy function, is the unique (up to scaling) $L^2$ solution to the stochastic Airy equation, a family of second order stochastic differential equations. Moreover, we obtain a coupling between the characteristic polynomial and a solution of the stochastic Airy equation which allows us to show that for any $\epsilon>0$, these two function are uniformly close by $N^{-1/6 + \epsilon}$ with overwhelming probability. These results build on the results of the authors in which the hyperbolic portion of the transfer matrix recurrence for the characteristic polynomial is analyzed.

Geometric and monotonic properties of hyper-Bessel functions

Abstract: Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler–Rayleigh inequalities and Laguerre–Polya class of real entire functions, intensively.

On zeros and growth of solutions of second order linear differential equations

Abstract: For a second order linear differential equation $f''+A(z)f'+B(z)f=0$, with $ A(z)$ and $B(z)$ being transcendental entire functions under some restriction, we have established that all non-trivial solutions are of infinite order. In addition, we have proved that these solutions have infinite number of zeros. Also, we have extended these results to higher order linear differential equations.

The many faces of the stochastic zeta function

Abstract: We introduce a framework to study the random entire function $\zeta_\beta$ whose zeros are given by the Sine$_\beta$ process, the bulk limit of beta ensembles. We present several equivalent characterizations, including an explicit power series representation built from Brownian motion. We study related distributions using stochastic differential equations. Our function is a uniform limit of characteristic polynomials in the circular beta ensemble; we give upper bounds on the rate of convergence. Most of our results are new even for classical values of $\beta$. We provide explicit moment formulas for $\zeta$ and its variants, and we show that the Borodin-Strahov moment formulas hold for all $\beta$ both in the limit and for circular beta ensembles. We show a uniqueness theorem for $\zeta$ in the Cartwright class, and deduce some product identities between conjugate values of $\beta$. The proofs rely on the structure of the Sine$_\beta$ operator to express $\zeta$ in terms of a regularized determinant.

Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

Abstract: We investigate a class of sharp Fourier extension inequalities on the planar curves s=|y|p, p>1. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if 1 4. In particular, this resolves the dichotomy of Jiang, Pausader, and Shao concerning the existence of extremizers for the Strichartz inequality for the fourth-order Schrodinger equation in one spatial dimension. One of our tools is a geometric comparison principle for n-fold convolutions of certain singular measures in ℝd, developed in the companion paper of Oliveira e Silva and Quilodran (Math. Proc. Cambridge Philos. Soc., (2019)). We further show that any extremizer exhibits fast L2-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves s=y|y|p−1, p>1.

Dynamical sets whose union with infinity is connected

Abstract: Suppose that $f$ is a transcendental entire function. In 2014, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected, and an example a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class. It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider's web. We use our results to give a large class of functions in the Eremenko-Lyubich class for which the escaping set is not a spider's web. Finally we give a novel topological criterion for certain sets to be a spider's web.

Generalized yang's conjecture on the periodicity of entire functions

Abstract: On the periodicity of transcendental entire functions, Yang’s Conjecture is proposed in [6, 13]. In the paper, we mainly consider and obtain partial results on a general version of Yang’s Conjecture, namely, if f(z)nf (k)(z) is a periodic function, then f(z) is also a periodic function. We also prove that if f(z)n+f (k)(z) is a periodic function with additional assumptions, then f(z) is also a periodic function, where n, k are positive integers.

Zeros of Gaussian Weyl-Heisenberg functions and hyperuniformity of charge.

Abstract: We study Gaussian random functions on the plane whose stochastics are invariant under the Weyl-Heisenberg group (twisted stationarity). We calculate the first intensity of their zero sets, both when considered as points on the plane, or as charges according to their phase winding. In the latter case, charges are always in a certain average equilibrium. We investigate the corresponding fluctuations, and show that in many cases they are suppressed at large scales (hyperuniformity). We also derive an asymptotic expression for the charge variance. Applications include poly-entire functions such as covariant derivatives of Gaussian entire functions, and zero sets of the short-time Fourier transform with general windows. We prove the following uncertainty principle: the expected number of zeros per unit area of the short-time Fourier transform of complex white noise is minimized, among all window functions, exactly by generalized Gaussians.

Inverse Sturm-Liouville problem with analytical functions in the boundary condition.

Abstract: The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

Algebrable sets of hypercyclic vectors for convolution operators

Abstract: We show that several convolution operators on the space of entire functions, such as the MacLane operator, support a dense hypercyclic algebra that is not finitely generated. Birkhoff’s operator also has this property on the space of complex-valued smooth functions on the real line.

Sampling of Entire Functions of Several Complex Variables on a Lattice and Multivariate Gabor Frames

Abstract: We give a general construction of entire functions in d complex variables that vanish on a lattice of the form Λ=A(Z+iZ)d for an invertible complex-valued matrix. As an application, we exhibit a cl...

Geometrically finite transcendental entire functions

Abstract: For polynomials, local connectivity of Julia sets is a much-studied and important property. Indeed, when the Julia set of a polynomial of degree $d\geq 2$ is locally connected, the topological dynamics can be completely described as a quotient of a much simpler system: angle $d$-tupling on the circle. For a transcendental entire function, local connectivity is less significant, but we may still ask for a description of the topological dynamics as the quotient of a simpler system. To this end, we introduce the notion of "docile" functions: a transcendental entire function with bounded postsingular set is docile if it is the quotient of a suitable disjoint-type function. Moreover, we prove docility for the large class of geometrically finite transcendental entire functions with bounded criticality on the Julia set. This can be seen as an analogue of the local connectivity of Julia sets for geometrically finite polynomials, first proved by Douady and Hubbard, and extends previous work of the second author and of Mihaljevic for more restrictive classes of entire functions.

Multiple zeta values for classical special functions

Abstract: We compute multiple zeta values (MZVs) built from the zeros of various entire functions, usually special functions with physical relevance. In the usual case, MZVs and their linear combinations are evaluated using a morphism between symmetric functions and multiple zeta values. We show that this technique can be extended to the zeros of any entire function, and as an illustration, we explicitly compute some MZVs based on the zeros of Bessel, Airy, and Kummer hypergeometric functions. We highlight several approaches to the theory of MZVs, such as exploiting the orthogonality of various polynomials and fully utilizing the Weierstrass representation of an entire function. On the way, an identity for Bernoulli numbers by Gessel and Viennot is revisited and generalized to Bessel–Bernoulli polynomials, and the classical Euler identity between the Bernoulli numbers and Riemann zeta function at even argument is extended to this same class.

On Julia limiting directions of meromorphic functions

Abstract: Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2π) is called a Julia limiting direction of f if there is an unbounded sequence {zn} in the Julia set J(f) satisfying limn→∞ arg zn = θ (mod 2π). We denote by L(f) the set of all Julia limiting directions of f. Our main result is that, for any non-empty compact set E ⊆ [0, 2π) and ρ ∈ [0, ∞], there are an entire function f of infinite lower order and a transcendental meromorphic function g of order ρ such that L(f) = L(g) = E. In addition, we have also constructed some transcendental entire functions whose lower order is ρ ∈ (1/2, ∞) and whose L(f) coincides with a certain kind of compact set. To prove our results, we have established a criterion for a direction θ to be a Julia limiting direction of a function by utilizing the growth rate of the function in the direction θ. The criterion may be of independent interest.

Lebesgue measure of escaping sets of entire functions

Abstract: For a transcendental entire function is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

Integrals of subharmonic functions and their differences with weight over small sets on a ray

Abstract: Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u ot\equiv -\infty$ and $v ot\equiv -\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E) $ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatoly A. Goldberg and Iossif V. Ostrovsky, and also generalizes analogs of the Edrei-Fuchs Lemma on small arcs for small intervals from the works of A. F. Grishin, M. L. Sodin, T. I. Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

On the Periodicity of Entire Functions

Abstract: The purpose of this paper is mainly to prove that if f is a transcendental entire function of hyper-order strictly less than 1 and $$f(z)^{n}+a_{1}f'(z)+\cdots +a_{k}f^{(k)}(z)$$ is a periodic function, then f(z) is also a periodic function, where n, k are positive integers, and $$a_{1},\cdots ,a_{k}$$ are constants. Meanwhile, we offer a partial answer to Yang’s Conjecture, theses results extend some previous related theorems.

Homotopy Hubbard Trees for Post-Singularly Finite Transcendental Entire Maps

Abstract: The main goal of this project is to investigate whether the concept of a Hubbard Tree, well established and widely used in polynomial dynamics, is also meaningful for transcendental entire functions. For a post-critically finite polynomial, its Hubbard Tree is the unique minimal embedded tree that contains all critical points and is forward invariant under the dynamics of the polynomial (and, in a certain sense, normalized on Fatou components). It is not difficult to adapt this definition to post-singularly finite (psf) transcendental entire maps. We show, however, that there are psf entire maps that do not admit a Hubbard Tree. The reason for this is the existence of asymptotic values. Partly in order to deal with that issue, we introduce the concept of a Homotopy Hubbard Tree. The essential difference to a Hubbard Tree is that a Homotopy Hubbard Tree is only required to be forward invariant up to homotopy relative to the post-singular set. Our main accomplishment in this work is to show that every psf transcendental entire map admits a Homotopy Hubbard Tree and that this tree is unique up to homotopy relative to the post-singular set. As a first step towards a classification of psf entire functions in terms of Homotopy Hubbard Trees, we show that a map is uniquely determined by its tree.

Thermodynamic formalism and integral means spectrum of logarithmic tracts for transcendental entire functions

Abstract: We provide an entirely new approach to the theory of thermodynamic formalism for entire functions of bounded type. The key point is that we introduce an integral means spectrum for logarithmic tracts which takes care of the fractal behavior of the boundary of the tract near infinity. It turns out that this spectrum behaves well as soon as the tracts have some sufficiently nice geometry which, for example, is the case for quasidisk, John or Hölder tracts. In these cases we get a good control of the corresponding transfer operators, leading to full thermodynamic formalism along with its applications such as exponential decay of correlations, central limit theorem and a Bowen’s formula for the Hausdorff dimension of radial Julia sets. This approach covers all entire functions for which thermodynamic formalism has been so far established and goes far beyond. It applies in particular to every hyperbolic function from any Eremenko-Lyubich analytic family of Speiser class S provided this family contains at least one function with Hölder tracts. The latter is, for example, the case if the family contains a Poincaré linearizer.

Speiser class Julia sets with dimension near one

Abstract: for any δ > 0 we construct an entire function f with three singular values whose Julia set has Hausdorff dimension at most 1 + δ. Stallard proved that the dimension must be strictly larger than 1 whenever f has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.