arXiv subjects

Edgar Saenz

Publications and source records attributed to Edgar Saenz.

4 recordsLinked to original sources

arXiv2026

A classification of bicritical dynamic portraits

The dynamic of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is determined by the forward orbits of its critical points. Such a map is called {\em postcritically finite} if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle. These orbits can be encoded in a finite directed graph called a {\em dynamic portrait}. In this work, we classify which abstract bicritical dynamic portraits of degree $d\geq2$ with at least 4 postcritical points are realizable exclusively by rational Thurston maps.

math.DS↗

arXiv2026

Sum of two strictly n-zero matrices

In this work we investigate when an $n\times n$ Jordan matrix can be written as the sum of two strictly $n$-zero matrices. In particular, we show that if $\mathbb{F}$ is an algebraically closed field of characteristic zero and $A$ is an $n\times n$ matrix over $\mathbb{F}$ with $\tr(A)=0$, then $A$ is the sum of two strictly $n$-zero matrices.

math.RA↗

arXiv2021

Realizing polynomial portraits

It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is realized as the ramification portrait of a postcritically finite polynomial, and classify which abstract polynomial portraits can only be realized by unobstructed maps.

math.DS↗

arXiv2016

Origami, affine maps, and complex dynamics

We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between NET maps and affine maps makes computation of many invariants tractable. In addition to this, NET maps are quite diverse, exhibiting many different behaviors. We discuss data, findings, and new phenomena.

math.DS↗