Related papers: H\'enon maps: a list of open problems
In this small paper we bring together various open problems on geometric multidimensional continued fractions.
We construct C-algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated to a matrix, the representation theory can be understood in terms of ``loop'' and…
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from…
Building on work of Doyle and Hyde on polynomial maps in one variable, we produce for each odd integer $d \geq 2$ a H\'enon map of degree $d$ defined over $\mathbb{Q}$ with at least $(d-4)^2$ integral periodic points. This provides a…
This is a survey on local dynamics of holomorphic maps in one and several complex variables, discussing in particular normal forms and the structure of local stable sets in the non-hyperbolic case, and including several proofs and a vast…
This note will discuss the dynamics of iterated cubic maps from the real or complex line to itself, and will describe the geography of the parameter space for such maps. It is a rough survey with few precise statements or proofs, and…
Stickiness is a well known phenomenon in which chaotic orbits expend an expressive amount of time in specific regions of the chaotic sea. This phenomenon becomes important when dealing with area-preserving open systems because, in this…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
The heterochaos baker maps are piecewise affine maps of the unit square or cube introduced in [Nonlinearity 34, 2021, 5744--5761], to provide a hands-on, elementary understanding of complicated phenomena in systems of large degrees of…
We discuss some challenging open problems in the geometric control theory and sub-Riemannian geometry.
We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new…
We make several new contributions to the study of proper holomorphic mappings between balls. Our results include a degree estimate for rational proper maps, a new gap phenomenon for convex families of arbitrary proper maps, and an…
We study the dynamics of a family of replicator maps, depending on two parameters. Such studies are motivated by the analysis of the dynamics of evolutionary games under selections. From the dynamics viewpoint, we prove the existence of…
We present a treasure trove of open problems in matrix and operator inequalities, of a functional analytic nature, and with various degrees of hardness.
We introduce the topological complexity of the work map associated to a robot system. In broad terms, this measures the complexity of any algorithm controlling, not just the motion of the configuration space of the given system, but the…
We show that special perturbations of a particular holomorphic map on $\mathbf{P}^k$ give us examples of maps that possess chaotic nonalgebraic attractors. Furthermore, we study the dynamics of the maps on the attractors. In particular, we…
This note presents an approach to studying the iterates of a mapping whose restriction to the complement of a finite set is continuous and open. The main examples to which the approach can be applied are piecewise monotone mappings defined…
Sometimes we obtain attractive results when associating facts to simple elements. The goal of this work is to introduce a possible alternative in the study of the dynamics of rational maps.
Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal…
In this note, we offer a palatable introduction to the field of arithmetic dynamics. That is, we study the patterns that arise when iterating a polynomial map. This note is accessible to those who have taken an introductory proof based…