Related papers: Dynamics Motivated by Sierpinski Fractals
We discuss the dynamic and structural properties of polynomial semigroups, a natural extension of iteration theory to random (walk) dynamics, where the semigroup $G$ of complex polynomials (under the operation of composition of functions)…
The Sierpinski Triangle (ST) is a fractal mathematical structure that has been used to explore the emergence of flat bands in lattices of different geometries and dimensions in condensed matter. Here we look into fractal features in the…
In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…
Many models of fractal growth patterns (like Diffusion Limited Aggregation and Dielectric Breakdown Models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we…
In this article, the dynamics of a one-parameter family of functions $f_{\lambda}(z) = \frac{\sin{z}}{z^2 + \lambda},$ $\lambda>0$, are studied. It shows the existence of parameters $0< \lambda_{1}< \lambda_{2}$ such that bifurcations occur…
Active fluids, composed of individual self-propelled agents, can generate complex large-scale coherent flows. A particularly important laboratory realization of such an active fluid is a system composed of microtubules, aligned in a…
Fluid-induced slip of fractures is characterized by strong multiphysics couplings. Three physical processes are considered: Flow, rock deformation and fracture deformation. The fractures are represented as lower-dimensional objects embedded…
We present a new model for two phase Darcy flows in fractured media, in which fractures are modelled as submanifolds of codimension one with respect to the surrounding domain (matrix). Fractures can act as drains or as barriers, since…
The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to…
Fractal lattices are self-similar structures with repeated patterns on different scales. As in other aperiodic lattices, the absence of translational symmetry can give rise to quantum localization effects. In contrast to low-dimensional…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves…
The iterative development process is a framework used to design products and applications across a wide range of domains. It centers around building prototypes, testing them, and updating based on the test results. We discuss how we applied…
Classical geometric fractals - Cantor set and Sierpinski continua - are presented in the manual as set-theoretic objects.
We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses…
Complex-dynamical fractal is a hierarchy of permanently, chaotically changing versions of system structure, obtained as the unreduced, causally probabilistic general solution of arbitrary interaction problem (physics/0305119,…
We study complex one-dimensional parameter slices in a three-parameter family of rational maps with two free critical points, obtained by imposing the existence of periodic orbits with prescribed multipliers. Using explicit…
Data series generated by complex systems exhibit fluctuations on many time scales and/or broad distributions of the values. In both equilibrium and non-equilibrium situations, the natural fluctuations are often found to follow a scaling…
We provide a natural canonical decomposition of postcritically finite rational maps with non-empty Fatou sets based on the topological structure of their Julia sets. The building blocks of this decomposition are maps where all Fatou…
In the first section we review recent results on the harmonic analysis of fractals generated by iterated function systems with emphasis on spectral duality. Classical harmonic analysis is typically based on groups whereas the fractals are…
This chapter describes how to use intersection and closest-hit shaders to implement real-time visualizations of complex fractals using distance functions. The Mandelbulb and Julia Sets are used as examples.