Related papers: Fractal dimensions of continuous piecewise linear …

Every quasi-attractor of an iterated function system (IFS) of continuous functions on a first-countable Hausdorff topological space is renderable by the probabilistic chaos game. By contrast, we prove that the backward minimality is a…

In this paper, we study cut sets of attractors of iteration function systems (IFS) in $\mathbb{R}^d$. Under natural conditions, we show that all irreducible cut sets of these attractors are perfect sets or single points. This leads to a…

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…

We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure…

We show that for the attractor $(K_{1},\dots,K_{q})$ of a graph directed iterated function system, for each $1\leq j\leq q$ and $\varepsilon>0$ there exits a self-similar set $K\subseteq K_{j}$ that satisfies the strong separation condition…

In the context of general iterated function systems (IFSs), we introduce bilinear fractal interpolants as the fixed points of certain Read-Bajraktarevi\'{c} operators. By exhibiting a generalized "taxi-cab" metric, we show that the graph of…

If a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters. For a piecewise linear two-parameter map we analyze…

The fast basin of an attractor of an iterated function system (IFS) is the set of points in the domain of the IFS whose orbits under the associated semigroup intersect the attractor. Fast basins can have non-integer dimension and comprise a…

We extend the hyperplane arrangement framework for neural network expressivity from the braid to discriminantal arrangements. Compatible piecewise linear functions are characterized by circuit relations and admit a matroidal description via…

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.

We compute the Hausdorff, upper box and packing dimensions for certain inhomogeneous Moran set constructions. These constructions are beyond the classical theory of iterated function systems, as different nonlinear contraction…

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool…

In this article, an iterated function system (IFS) is considered on the real projective line $\mathbb{RP}^1$ so that the attractor is a Cantor-like set. Hausdorff dimension of this attractor is estimated. The existence of a probability…

We introduce a technique that uses projection properties of fractal percolation to establish dimension conservation results for sections of deterministic self-similar sets. For example, let $K$ be a self-similar subset of $\mathbb{R}^2$…

We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$…

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 provide a rigorous study on dimensions of fractal interpolation function defined on a closed and bounded interval of $\mathbb{R}$ which is associated to a continuous function with respect to a base function, scaling functions and a…

In this paper we will introduce the methodology of analysis of the convex hull of the attractors of iterated functional systems (IFS) - compact fixed sets of self-similarity mapping. The method is based on a function which for a direction,…

In this paper, we will consider subfractals of hyperbolic iterated function systems which satisfy the open set condition. The subfractals will consist of points associated with infinite strings from a subshift of finite type or sofic…

We consider a class of planar self-affine sets which we call "box-like". A box-like self-affine set is the attractor of an iterated function system (IFS) of affine maps where the image of the unit square, [0,1]^2, under arbitrary…