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Adi Glücksam

Publications and source records attributed to Adi Glücksam.

12 recordsLinked to original sources
arXiv2026

A Comparison Test for Meromorphic Extensions

We provide a comparison test for meromorphic extensions, i.e., if two series are ``close enough" then the existence of a meromorphic extension of one to the entire complex plane ensures a similar extension for the other. We use this result to generate new examples of Dirichlet series admitting meromorphic extensions. Moreover, we demonstrate that our requirements are optimal by constructing a collection of counterexamples where the series are ``close but not enough": one series admits a meromorphic extension while the other possesses a natural boundary.

math.CV↗
arXiv2025

Simply Connected Wandering Domains of Small Order Entire Functions

We show that any bounded, simply connected domain with analytic boundary can be realised as a wandering domain of an entire function of any prescribed order in $(0, 1)$. Extending results of Boc Thaler, our construction simultaneously prescribes the domain and the exact order of the map. In particular, we produce the first examples of entire functions with bounded simply connected wandering domains of each order in $(0,1/2).$

math.CV↗
arXiv2025

Analysis of Boundary Behaviour of Quasidisks and Jordan Repellers

We investigate the fine properties of harmonic measure and boundary rotation. By focusing on quasidisks and, in particular, on connected Jordan Repellers arising from conformal expanding dynamical systems, we explore those using the deep interplay between geometric function theory and dynamical systems as a unified framework. While some of the results presented here are regarded as 'folklore' among experts, they lack rigorous proofs in the existing literature. We fill this gap by providing a comprehensive, referable treatment using a novel approach that also expands existing results.

math.CV↗
arXiv2025

Measurable entire functions II

Let $\mathcal{E}$ denote the space of entire functions with the topology of uniform convergence on compact sets. The action of $\mathbb C$ by translations on $\mathcal E$ is defined by $T_zf(w) = f(w+z)$. Let $\mathcal{U}$ denote the set of entire functions whose orbit under $T$ is dense. Birkhoff showed, in [B], that $\mathcal{U}$ is not empty. One of the problems in the collection by T-C Dinh and N. Sibony [DS] asks whether there exists an invariant probability measure on $\mathcal{E}$ whose support is contained in $\mathcal U$. We will show how an old construction of the second author can be modified to provide a positive answer to their question. Furthermore, we modify the construction to produce a wealth of ergodic measures on the space of entire functions of several complex variables.

math.DS↗
arXiv2025

On the (Dis)connection Between Growth and Primitive Periodic Points

In 1972, Cornalba and Shiffman showed that the number of zeros of an order zero holomorphic function in two or more variables can grow arbitrarily fast. We generalize this finding to the setting of complex dynamics, establishing that the number of isolated primitive periodic points of an order zero holomorphic function in two or more variables can grow arbitrarily fast as well. This answers a recent question posed by L. Buhovsky, I. Polterovich, L. Polterovich, E. Shelukhin and V. Stojisavljević.

math.CV↗
arXiv2023

An approximate solution to Erdös' maximum modulus points problem

In this note we investigate the asymptotic behavior of the number of maximum modulus points, of an entire function, sitting in a disc of radius $r$. In 1964, Erd\Humlaut{o}s asked whether there exists a non-monomial function so that this quantity is unbounded? tends to infinity? In 1968 Herzog and Piranian constructed an entire map for which it is unbounded. Nevertheless, it is still unknown today whether it is possible that it tends to infinity or not. In this paper, we construct a transcendental entire function that is arbitrarily close to satisfying this property, thereby giving the strongest evidence supporting a positive answer to this question.

math.CV↗
arXiv2023

Unbounded fast escaping wandering domains

We introduce a new approximation technique into the context of complex dynamics that allows us to construct examples of transcendental entire functions with unbounded wandering domains. We provide examples of entire functions with an orbit of unbounded fast escaping wandering domains, answering a long-standing question of Rippon and Stallard. Moreover, these examples cover all possible types of simply connected wandering domains in terms of convergence to the boundary. In relation to a conjecture of Baker, it was unknown whether functions of order less than one could have unbounded wandering domains. For any given order greater than $1/2$ and smaller than $1$, we provide an entire function of such order with an unbounded wandering domain.

math.DS↗
arXiv2023

Frequently oscillating families related to subharmonic functions

The goal of this note is to extend the result bounding from bellow the minimal possible growth of frequently oscillating subharmonic functions to a larger class of functions that carry similar properties. We refine and find further applications for the technique presented by Jones and Makarov in their celebrated paper, Density properties of harmonic measure.

math.CV↗
arXiv2020

Harmonic measure of the outer boundary of colander sets

We present two companion results: Phragmén-Lindelöf type tight bounds on the minimal possible growth of subharmonic functions with recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer boundary of colander sets.

math.CV↗
arXiv2018

Measurably entire functions and their growth

In 1997 B. Weiss introduced the notion of measurably entire functions and proved that they exist on every arbitrary free C- action defined on standard probability space. In the same paper he asked about the minimal possible growth of measurably entire functions. In this work we show that for every arbitrary free C- action defined on a standard probability space there exists a measurably entire function whose growth does not exceed exp (exp[log^p |z|]) for any p > 3. This complements a recent result by Buhovski, Glücksam, Logunov, and Sodin (arXiv:1703.08101) who showed that such functions cannot grow slower than exp (exp[log^p |z|]) for any p < 2.

math.DS↗
arXiv2013

Ergodic Multiplier Properties

In this article we will see some properties that guarantee that a product of an ergodic non-singular action and a probability preserving ergodic action is also an ergodic action. We will start by proving 'The multiplier theorem' for locally compact abelian groups. Then we will show that for certain locally compact Polish groups (Moore groups, and minimally weakly mixing groups), a non-singular G action is weakly mixing if and only if any finite dimensional G-invariant subspace of L_\infty is trivial. Finally, we will show that the Gaussian action associated to the infinite dimensional irreducible representation of the continuous Heisenberg group, H_3, is weakly mixing but not mildly mixing.

math.DS↗