Related papers: Dynamics of Newton maps

A "Littlewood polynomial" is a polynomial whose coefficients are all 1 or -1. The set of all complex roots of all Littlewood polynomials exhibits many complicated, beautiful and fascinating patterns. Some fractal regions of this set closely…

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter $\lambda$, and generalize this characterization to cubic real polynomial maps,…

We use the stabilization functors to study the combinatorial aspects of the $F$-polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for…

We present sufficient conditions so that a conformal map between planar domains whose boundary components are Jordan curves or points has a continuous or homeomorphic extension to the closures of the domains. Our conditions involve the…

We review the Shimura-Taniyama method for computing the Newton polygon of an abelian variety with complex multiplication. We apply this method to cyclic covers of the projective line branched at three points. As an application, we produce…

We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which…

We study the problem of persistence of attractors with smooth boundary for a class of set-valued dynamical systems that naturally arise in the context of random and control dynamical systems, as well as in systems modeling the dynamical…

Let J and J' be Jordan rings. We prove under some conditions that if J contains a nontrivial idempotent, then n-multiplicative maps and n-multiplicative derivations from J to J' are additive maps.

In our previous works we proved a finiteness property of the radii of convergence functions associated with a vector bundle with connection on $p$-adic analytic curves. We showed that the radii are locally constant functions outside a…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

Let $f$ be a polynomial automorphism of the affine plane. In this paper we consider the possibility for it to possess infinitely many periodic points on an algebraic curve $C$. We conjecture that this happens if and only if $f$ admits a…

In 2004 Oort studied the foliation on the space of $p$-divisible groups. In his theory, special leaves called central streams play an important role. It is still meaningful to investigate central streams, for example, there remain a lot of…

Discrete models have a long tradition in engineering, including finite state machines, Boolean networks, Petri nets, and agent-based models. Of particular importance is the question of how the model structure constrains its dynamics. This…

This paper is an extension of an earlier paper that dealt with global dynamics in autonomous triangular maps. In the current paper, we extend the results on global dynamics of autonomous triangular maps to periodic non-autonomous triangular…

We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed…

We study expanding circle maps interacting in a heterogeneous random network. Heterogeneity means that some nodes in the network are massively connected, while the remaining nodes are only poorly connected. We provide a probabilistic…

The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer…

To directed graphs with unique sink and source we associate a noncommutative associative alsgebra and a polynomial over this algebra. Edges of the graph correspond to pseudo-roots of the polynomial. We give a sufficient condition when…

We prove that every transcendental meromorphic map f with a disconnected Julia set has a weakly repelling fixed point. This implies that the Julia set of Newton's method for finding zeroes of an entire map is connected. Moreover, extending…

We prove that, for every invertible horizontal-like map (i.e., H{\'e}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after…