Related papers: Iterated function systems over arbitrary shift spa…

We investigate whether the Hutchinson operator associated with the iterated function system (IFS) is continuous. It clarifies several partial results scattered across recent literature. While the main example for IFS with strict attractor…

Let $M$ be a smooth ($C^{\infty}$) manifold, $F_1,...,F_n$ be vector fields on $M$ generating the corresponding flows $\Phi_1,...,\Phi_n$, and $\alpha_1,...,\alpha_{n}:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$…

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures;…

We extend classical basis constructions from Fourier analysis to attractors for affine iterated function systems (IFSs). This is of interest since these attractors have fractal features, e.g., measures with fractal scaling dimension.…

We try to define the more general form of iterative processes in which the Pomeau-Manneville and the Feigenbaum scenario may occur along with their specific scaling properties. Doing this we need to generalize other basic concepts. Thus,…

The control system described by Urysohn type integral equation is considered where the system is nonlinear with respect to the phase vector and is affine with respect to the control vector. The control functions are chosen from the closed…

We study random independent and identically distributed iterations of functions from an iterated function system of homeomorphisms on the circle which is minimal. We show how such systems can be analyzed in terms of iterated function…

Intermittent dynamics is characterized by long periods of different types of dynamical characteristics, for instance almost periodic dynamics alternated by chaotic dynamics. Critical intermittency is intermittent dynamics that can occur in…

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called the automorphism orbits of $G$, and their number is denoted by $\omega(G)$. Let $\mathbb{F}$ be an infinite field, and let $UT_n(\mathbb{F})$ denote the…

We apply some methods and technique of complex dynamics to study the set of symmetries of attractors of holomorphic Iterated Function Systems (IFS), as well as relations between IFS sharing the same attractor.

Let $(X, \{w_j \}_{j=1}^m, \{p_j \}_{j=1}^m)$ ($2 \leq m < \infty$) be a contractive iterated function system (IFS), where $X$ is a compact subset of ${\Bbb{R}}^d$. It is well known that there exists a unique nonempty compact set $K$ such…

This paper is in the form of an essay. It defines fractal tops and code space structures associated with set-attractors of hyperbolic iterated function systems (IFSs). The fractal top of an IFS is associated with a certain shift invariant…

This paper introduces an intrinsic theory of Thermodynamic Formalism for Iterated Functions Systems with general positive continuous weights (IFSw).We study the spectral properties of the Transfer and Markov operators and one of our first…

We define iteration of functions that map n-dimensional vector spaces into m-dimensional vector spaces (m at most equal to n). It happens that usual iteration and Fibonacci iterative methods become special cases of this generalized…

For any primitive proper substitution \sigma, we give explicit constructions of countably many pairwise non-isomorphic substitution dynamical systems {(X_{\zeta_n}, T_{\zeta_n})}_{n=1}^{\infty} such that they all are (strong) orbit…

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…

This paper introduces Fourier duality for a class of affine iterated function systems (IFS) T_i. These systems are determined by a finite family of contractive affine maps in R^d. Our Fourier duality applies to the resulting probability…

For any self-similar measure $\mu$ in $\mathbb{R}$, we show that the distribution of $\mu$ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the IFS. This generalizes the net…

The goal of this note is to generalize Thurston's Topological Characterization of Rational Functions to the setting when both the covering degree and the set of marked points are infinite. A relevant class of branched coverings are…

Iterated Graph Systems (IGS) transplant ideas from fractal geometry into graph theory. Building on this framework, we extend Edge IGS from the primitive to the reducible setting. Within this broader context, we formulate rigorous…