Related papers: On fractal Peano curves

This paper concerns the local connectedness of components of self-similar sets. Given an equal partition of the unit square into n*n small squares, we may choose arbitrarily two or more of them and form an iterated function system. The…

We prove that every Peano continuum with uncountably many local cut points is a topological fractal. This extends some recent results and it partially answers a conjecture by Hata. We also discuss the number of mappings which are necessary…

While self-similar sets have no tangents at any single point, self-affine curves can be smooth. We consider plane self-affine curves without double points and with two pieces. There is an open subset of parameter space for which the curve…

In this paper, we present an effective method to characterize completely when a disconnected fractal square has only finitely many connected components. Our method is to establish some graph structures on fractal squares to reveal the…

We refer here to the surprising construction made by Giuseppe Peano in 1890. He gave an example of a continuous function (called now the Peano curve) from the unit interval to the whole unit square. We show here the existence of a more…

We prove that every usco multimap $\Phi:X\to Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_\sigma$-measurable selection. On the other hand, for the split interval $\ddot{\mathbb I}$ and the projection $P:\ddot{\mathbb…

It is proved that for any mapping of a unit segment to a unit square, there is a pair of points of the segment for which the square of the Euclidean distance between their images exceeds the distance between them on the segment by at least…

Let F be a subfield of a commutative field extending R. Let \phi_2: F^2 \times F^2 \to F, \phi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 \to F^2 preserves distance d \geq 0 if for each x,y \in R^2 |x-y|=d implies…

Let \phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 -> C^2 preserves distance d>=0 if for each x,y \in R^2 \phi(x,y)=d^2 implies \phi(f(x),f(y))=d^2. We prove that if x,y \in R^2 and |x-y|=(2\sqrt{2}/3)^k \cdot…

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…

We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is…

Given a multi-valued function $\Phi$ on a topological space $X$ we study the properties of its fixed fractal, which is defined as the closure of the orbit $\Phi^\omega(Fix(\Phi))=\bigcup_{n\in\omega}\Phi^n(Fix(\Phi))$ of the set…

Local iterated function systems are an important generalisation of the standard (global) iterated function systems (IFSs). For a particular class of mappings, their fixed points are the graphs of local fractal functions and these functions…

We construct Peano curves $\gamma : [0,\infty) \to \mathbb{R}^2$ whose "footprints" $\gamma([0,t])$, $t>0$, have $C^\infty$ boundaries and are tangent to a common continuous line field on the punctured plane $\mathbb{R}^2 \setminus…

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a…

Let $f \colon X \rightarrow Y$ be a resolvable-measurable mapping of a metrizable space $X$ to a regular space $Y$. Then $f$ is piecewise continuous. Additionally, for a metrizable completely Baire space $X$, it is proved that $f$ is…

Let a>0, F: R^2 -> R^2 be a differentiable (not necessarily C^1) map and Spec(F) be the set of (complex) eigenvalues of the derivative F'(p) when p varies in R^2. (a) If Spec(F) is disjoint of the interval [1,1+a[, then Fix(F) has at most…

A new calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called…

This paper introduces the concept of Fractal Frenet equations, a set of differential equations used to describe the behavior of vectors along fractal curves. The study explores the analogue of arc length for fractal curves, providing a…

We show that for every $n \in \mathbb N$ there is a collection of points $p_1, \ldots, p_n$ and lines $\ell_1, \ldots, \ell_n$ in the unit square such that for any $i$ we have $p_i \in \ell_i$ and the distance from $p_i$ to any other line…