Related papers: Blowups and Tops of Overlapping Iterated Function …
The theory of fractal tilings of fractal blow-ups is extended to graph-directed iterated function systems, resulting in generalizations and extensions of some of the theory of Anderson and Putnam and of Bellisard et al. regarding…
New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our…
The top of the attractor $A$ of a hyperbolic iterated function system $\left\{ f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}|i=1,2,\dots,M\right\} $ is defined and used to extend self-similar tilings to overlapping systems. The theory…
Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal…
In 1996, Strichartz introduced reverse iterated function systems (RIFS) $\mathcal{F}=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ of expanding mappings on $\mathbb{Z}$ and left the determination of the general dimension formulas of invariant sets as an…
We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a…
This preliminary paper presents initial explorations in rendering Iterated Function System (IFS) fractals using a differentiable rendering pipeline. Differentiable rendering is a recent innovation at the intersection of computer graphics…
The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence…
A simple, yet unifying method is provided for the construction of tilings by tiles obtained from the attractor of an iterated function system (IFS). Many examples appearing in the literature in ad hoc ways, as well as new examples, can be…
This paper describes the celebrated aperiodic hat tiling by Smith et al. [Comb. Theory 8 (2024), 6] as generated by an overlapping iterated function system. We briefly introduce and study infinite sequences of iterated function systems that…
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…
Following the work of Louisa and Michael Barnsley on results in tops of iterated function systems, we extend their work to graph-directed iterated function systems by investigating the relationship between top addresses and shift spaces.…
Several important conjectures in Fractal Geometry can be summarised as follows: If the dimension of a self-similar measure in $\mathbb{R}$ does not equal its expected value, then the underlying iterated function system contains an exact…
We study new relations between countable iterated function systems (IFS) with overlaps, Smale endomorphisms and random systems with complete connections. We prove that stationary measures for countable conformal IFS with overlaps and…
Fulton and MacPherson famously constructed a configuration space that encodes infinitesimal collision data by blowing up the diagonals. We observe that when generalizing their approach to configuration spaces of filtered manifolds (e.g. jet…
Until recently, it was an important open problem in Fractal Geometry to determine whether there exists an iterated function system acting on $\mathbb{R}$ with no exact overlaps for which cylinders are super-exponentially close at all small…
Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts…
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…
This article discusses the notion of convergence of sequences of iterated function systems. The technique of iterated function systems is one of the several methods to construct objects with fractal nature, and the fractals obtained with…
A powerful statistical interpolating concept, which we call \emph{fully lifted} (fl), is introduced and presented while establishing a connection between bilinearly indexed random processes and their corresponding fully decoupled (linearly…