Related papers: Symbolic template iterations of complex quadratic …

We formulate general rules for a coarse-graining of the dynamics, which we term `symbolic dynamics', of feedback networks with monotone interactions, such as most biological modules. Networks which are more complex than simple cyclic…

The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic…

For certain typical perturbations $(f_n)_n$ of a rational map $f$ with parabolic cycles, we investigate the relations between the Hausdorff convergence of Julia sets and invariant rays, and the horocyclic convergence of multipliers of…

In this work, a symbolic dynamical formulation based upon discrete iterative mappings derived from the Collatz conjecture is introduced. It is demonstrated that this formulation naturally induces a ternary alphabet useful for characterizing…

In this article, we attempt to study the possible link between the dynamics of a circle map and the caustics of its iterations. The attention is on a geometrically defined off-center reflections, which, coincidentally, is also a…

This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general…

Living systems exhibit complex yet organized behavior on multiple spatiotemporal scales. To investigate the nature of multiscale coordination in living systems, one needs a meaningful and systematic way to quantify the complex dynamics, a…

Periodic orbits and cycles, respectively, play a significant role in discrete- and continuous-time dynamical systems (i.e. maps and flows). To succinctly describe their shifts when the system is applied perturbation, the notions of…

We argue that discrete dynamics has natural links to the theory of analytic functions. Most important, bifurcations and chaotic dynamical properties are related to intersections of algebraic varieties. This paves the way to identification…

In the early 1980's Thurston gave a topological characterization of rational maps whose critical points have finite iterated orbits (\cite{Th,DH1}): given a topological branched covering $F$ of the two sphere with finite critical orbits, if…

Template tasks have emerged as a clean testbed for asking whether transformers reason with abstract symbols rather than concrete token names. We study the fixed-label classification version of this problem, where train and test examples…

We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally…

Recent progress of symbolic dynamics of one- and especially two-dimensional maps has enabled us to construct symbolic dynamics for systems of ordinary differential equations (ODEs). Numerical study under the guidance of symbolic dynamics is…

We show how machine learning methods can unveil the fractional and delayed nature of discrete dynamical systems. In particular, we study the case of the fractional delayed logistic map. We show that given a trajectory, we can detect if it…

The increased availability of interactive maps on the Internet and on personal mobile devices has created new challenges in computational cartography and, in particular, for label placement in maps. Operations like rotation, zoom, and…

The iteration of rational maps is well-understood in dimension 1 but less so in higher dimensions. We study some maps on spaces of matrices which present a weak complexity with respect to the ring structure. First we give some properties of…

The Fatou-Julia decomposition is significant in the study of iterations of holomorphic mappings. Such a decomposition can be also considered for foliations in a unified manner. Although the decomposition will be fundamental in the study, it…

Template 3D shapes are useful for many tasks in graphics and vision, including fitting observation data, analyzing shape collections, and transferring shape attributes. Because of the variety of geometry and topology of real-world shapes,…

We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single…