Related papers: On the Space of Iterated Function Systems and Thei…

In a pedagogical but exhaustive manner, this survey reviews the main results on input-to-state stability (ISS) for infinite-dimensional systems. This property allows estimating the impact of inputs and initial conditions on both the…

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$…

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the…

Given a nondegenerate compact perfect and Hausdorff topological space $X$,$n\in \mathbb{N}$ and a function $f:X\rightarrow X$, we consider the $n$-fold symmetric product of $X$, $F_n(X)$ and the induced function $F_n(f):F_n(X)\rightarrow…

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…

For a given compact Hausdorff space $X$, we construct the space $OS_{f}(X)$ of normed, order-preserving, weakly additive, positively homogeneous and semi-additive functionals (for brevity, semi-additive functionals) and it is proved that…

Let $C(\mathbf I)$ be the set of all continuous self-maps from ${\mathbf I}=[0,1]$ with the topology of uniformly convergence. A map $f\in C({\mathbf I})$ is called a transitive map if for every pair of non-empty open sets $U,V$ in…

We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are…

Empirically defining some constant probabilistic orbits of f(x) and g(x) iterated high-order functions, the stability of these functions in possible entangled interaction dynamics of the environment through its orbit's connectivity (open…

In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that…

The sub-additive pressure function $P(s)$ for an affine iterated function system (IFS) and the affinity dimension, defined as the unique solution $s_0$ to $P(s_0)=1$, were introduced by K. Falconer in his seminal 1988 paper on self-affine…

Motivated by the problem of dealing with incomplete or imprecise acquisition of data in computer vision and computer graphics, we extend results concerning the stability of persistent homology with respect to function perturbations to…

The notion of stochastic shadowing property is introduced. Relations to stochastic stability and standard shadowing are studied. Using tent map as an example it is proved that, in contrast to what happens for standard shadowing, there are…

We consider iterated function systems (IFS) in ${\mathbb R}^d$ for $d\ge 3$ of the form $\{f_j(x) = \lambda {\mathcal O} x + a_j\}_{j=0}^m$, with $a_0=0$ and $m\ge 1$. Here $\lambda\in (0,1)$ is the contraction ratio and ${\mathcal O}$ is…

Consider two objects associated to the Iterated Function System (IFS) $\{1+\lambda z,-1+\lambda z\}$: the locus $\mathcal{M}$ of parameters $\lambda\in\mathbb{D}\setminus\{0\}$ for which the corresponding attractor is connected; and the…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…

Under suitable conditions, with respect to some property on a random iterated function system(RIFS), it is shown how the system satisfies in this property almost surely.

We study conformal iterated function systems (IFS) $\mathcal S = \{\phi_i\}_{i \in I}$ with arbitrary overlaps, and measures $\mu$ on limit sets $\Lambda$, which are projections of equilibrium measures $\hat \mu$ with respect to a certain…

We study the properties of topological spaces $(X,\tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ on $X$ has a basis that is (uniformly) definable. Examples of such spaces include the canonical…

This paper investigates spaces equipped with a family of metric-like functions satisfying certain axioms. These functions provide a unified framework for defining topology, uniformity, and diffeology. The framework is based on a family of…