Related papers: Deautonomisation by singularity confinement and de…
In this paper, we develop several tools to study the degree growth and stabilization of monomial maps. Using these tools, we can classify semisimple three dimensional monomial maps by their dynamical behavior.
We study an atmospheric column and its discretization. Because of numerical considerations, the column must be divided into two parts: (1) a surface layer, excluded from the computational domain and parameterized, and (2) the rest of the…
We consider a dynamics of a generic birational plane map \Phi_n: CP^2 \to CP^2, CP^2 -image of the birational mapping (inverse map is also rational)F_n : C^2 \to C^2 and its such important characteristic as the Arnold complexity C_A(k),…
We study the plane automorphisms given by polynomials with certain degree decompositions.
A model of discrete spacetime on a microscopic level is considered. It is a directed acyclic dyadic graph. This is the particular case of a causal set. The goal of this model is to describe particles as some repetitive symmetrical…
Dual maps have been introduced as a generalization to higher dimensions of word substitutions and free group morphisms. In this paper, we study the action of these dual maps on particular discrete planes and surfaces -- namely stepped…
In the past many papers have appeared which simulated surface growth with different growth models. The results showed that, if models differed only slightly in their `growth' rules, the resulting surfaces may belong to different…
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…
We study nonautonomous mappings of the plane by means of spaces of initial conditions. First we introduce the notion of a space of initial conditions for nonautonomous systems and we study the basic properties of general equations that have…
Many growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting…
The dynamics of a one dimensional growth model involving attachment and detachment of particles is studied in the presence of a localized growth inhomogeneity along with anchored boundary conditions. At large times, the latter enforce an…
In this paper we investigate the integrability of two-dimensional partial difference equations using the newly developed techniques of study of the degree of the iterates. We show that while for generic, nonintegrable equations, the degree…
We study canonical heights for plane polynomial mappings of small topological degree. In particular, we prove that for points of canonical height zero, the arithmetic degree is bounded by the topological degree and hence strictly smaller…
A one-dimensional cellular automaton with a probabilistic evolution rule can generate stochastic surface growth in $(1 + 1)$ dimensions. Two such discrete models of surface growth are constructed from a probabilistic cellular automaton…
We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
I show that the evolution of a two dimensional surface in a Laplacian field can be described by Hamiltonian dynamics. First the growing region is mapped conformally to the interior of the unit circle, creating in the process a set of…
The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational…
Automata networks can be seen as bare finite dynamical systems, but their growing theory has shown the importance of the underlying communication graph of such networks. This paper tackles the question of what dynamics can be realized up to…
We examine a family of discrete second-order systems which are integrable through reduction to a linear system. These systems were previously identified using the singularity confinement criterion. Here we analyse them using the more…