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Adrián Esparza-Amador

Publications and source records attributed to Adrián Esparza-Amador.

6 recordsLinked to original sources
arXiv2026

Dynamics of Newton's method for odd and even elliptic functions

We investigate Newton's method applied to any odd or any even elliptic function with an arbitrary period lattice. For any function of this type whose set of poles coincides with its period lattice, we show that the Julia set of its Newton map is connected, as long as no Herman rings exist. Moreover, we provide sufficient conditions on the Newton's method of any odd or even elliptic function to exhibit wandering domains coexisting with attracting basins. This phenomenon was first reported by Florido and Fagella for the Newton's method applied to a family of entire functions; however, our approach does not employ the lifting technique. We also provide a detailed study of a one-parameter family of elliptic functions given by $\wp_Λ+b$ with $Λ$ any triangular period lattice and $b\in\mathbb{C}$. We show that their associated Newton maps do not exhibit Herman rings or Baker domains, and any other Fatou component, including wandering, is bounded.

math.DS↗
arXiv2024

Thermodynamic formalism and hyperbolic Baker domains: Real-analyticity of the Hausdorff dimension

We consider the family of entire maps given by $f_{\ell,c}(z)=\ell+c-(\ell-1)\log c-e^z$, where $c\in D(\ell,1)$ and $\ell\in\mathbb N$, $\ell\geq2$. By using the property of $f_{\ell,c}$ to be dynamically projected to an infinite cylinder $\mathbb C/2πI\mathbb Z$, where the thermodynamic formalism tools are well-defined, we prove as a main result on this work, the real-analyticity of the map $c\mapsto HD(\mathcal{J}_r(f_{\ell,c}))$, here $\mathcal{J}_r(f_{\ell,c})$ is the radial Julia set.

math.DS↗
arXiv2023

Thermodynamic formalism for entire transcendental maps with hyperbolic Baker Domains

We provide a version of a thermodynamic formalism of entire transcendental maps that exhibit Baker domains, denoted as $f_{\ell, c}: \mathbb C\to \mathbb C$ and defined by $f_{\ell, c}(z)= c-(\ell-1)\log c+ \ell z- e^z$, where $\ell \in \mathbb N$, with $\ell \geq 2 $ and $c$ belongs to the disk $ D(\ell, 1)$ in the complex plane. We show in particular the existence and uniqueness of conformal measures and that the Hausdorff dimension is the unique zero of the pressure function $t\to P(t)$, for $t>1,$ where $J_r(f)$ is the radial Julia set.

math.DS↗
arXiv2021

Commuting functions in class K and families of wandering Baker domains

Given commuting functions f,g, with at most a countable compact set of essential singularities, recent results for entire functions are extended to prove that Julia sets match, J(f)=J(g), in a particular case in the class K. With this result on hand, a general form of functions in this class, is constructed with wandering Baker domains.

math.DS↗
arXiv2020

Elementary characters on semigroups: the rational case

Since polynomials form a subsemigroup of the semigroup of rational functions, every character on rational functions is a character on polynomials. On the other direction, not every character on polynomials is the restriction of a character on rational functions. What are the characters on polynomials that can be extended to rational functions? In this work, we conjecture that the only characters that can be extended are those that depends on the degree, often called elementary. Also, we construct two example of character on polynomials, not elementaries, that cannot be extended to rational functions.

math.RA↗