Related papers: Iterated function systems, Ruelle operators, and i…
We introduce the novel concept of hypercomplex iterated function system (IFS) on the complete metric space $(\mathbb{A}_{n+1}^k,d)$ and define its hypercomplex attractor. Systems of hypercomplex function systems arising from hypercomplex…
We discuss the problem of bounding the Fourier transforms of stationary measures of iterated function systems (IFSs) and how the pseudo-randomness of the IFS either due to arithmetic, algebraic or geometric reasons is reflected in the…
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…
For a Borel probability measure $\mu$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(\mu)$ admits an orthogonal basis of exponential functions. In this paper, we study the spectrality of fractal measures…
Measures generated by Iterated Function Systems composed of uncountably many one--dimensional affine maps are studied. We present numerical techniques as well as rigorous results that establish whether these measures are absolutely or…
The orbit of a point $x\in X$ in a classical iterated function system (IFS) can be defined as $\{f_u(x)=f_{u_n}\circ\cdots \circ f_{u_1}(x):$ $u=u_1\cdots u_n$ is a word of a full shift $\Sigma$ on finite symbols and $f_{u_i}$ is a…
Iterated function systems (IFS) provide a powerful method for constructing fractals and modeling complex structures. This paper develops the notion of a dynamical system of IFS to study how an initial IFS evolves over time. We construct a…
Within the study of uncertain dynamical systems, iterated random functions are a key tool. There, one samples a family of functions according to a stationary distribution. Here, we introduce an extension, where one sample functions…
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…
Suppose $\{f_1,...,f_m\}$ is a set of Lipschitz maps of $\mathbb{R}^d$. We form the iterated function system (IFS) by independently choosing the maps so that the map $f_i$ is chosen with probability $p_i$ ($\sum_{i=1}^m p_i=1$). We assume…
In this paper we consider Iterated Function Systems (IFS) on the real line consisting of continuous piecewise linear functions. We assume some bounds on the contraction ratios of the functions, but we do not assume any separation condition.…
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
We study contraction conditions for an iterated function system of continuous maps on a metric space which are chosen randomly, identically and independently. We investigate metric changes, preserving the topological structure of the space,…
We prove that the idempotent Markov operator generated by contractive max plus normalized iterated function system (IFS) is also a contractive map w.r.t. natural metrics on the space of idempotent measures. This gives alternative proofs of…
This paper introduces a new class of iterated function systems (IFSs) called R-IFSs, which include both rotation/reflection maps and contraction maps. The study of R-IFSs is motivated by the recent research direction on enriching IFSs by…
We consider iterated function systems $\mathrm{IFS}(T_1,\dots,T_k)$ consisting of continuous self maps of a compact metric space $X$. We introduce the subset $S_{\mathrm{t}}$ of {\emph{weakly hyperbolic sequences}} $\xi=\xi_0\ldots\xi_n…
Quantum Iterated Function System on a complex projective space is defined by a family of linear operators on a complex Hilbert space. The operators define both the maps and their probabilities by one algebraic formula. Examples with…
Let $d$ be a positive integer, and let $\mu$ be a finite measure on $\br^d$. In this paper we ask when it is possible to find a subset $\Lambda$ in $\br^d$ such that the corresponding complex exponential functions $e_\lambda$ indexed by…
This paper contains four main results associated with an attractor of a projective iterated function system (IFS). The first theorem characterizes when a projective IFS has an attractor which avoids a hyperplane. The second theorem…
We study invariant measures for random countable (finite or infinite) conformal iterated function systems (IFS) with arbitrary overlaps. We do not assume any type of separation condition. We prove, under a mild assumption of finite entropy,…