surftop5.pdf

2008

In this article we classified the points of the well-known fractal set Sierpinski Gasket (SG) according to their addresses. We also gave a characterization of points of SG that describes the relation between their addresses and their components in R 2 .

Proceedings of the American Mathematical Society, 2005

For self-similar sets, the existence of a feasible open set is a natural separation condition which expresses geometric as well as measure-theoretic properties. We give a constructive approach by defining a central open set and characterizing those points which do not belong to feasible open sets.

Fractal and Fractional

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension D which exceeds the topological dimension d. In this regard, we point out that the constitutive inequality D>d can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined.

Probability Theory and Related Fields, 1987

arXiv (Cornell University), 2017

We observed and described the generalized Sierpiński Arrowhead Curve in our previous paper [K17a]. Now we focus on its background structure. In Section 1 we summarize our previous results on the triangular grid and supplement them with Hamiltonian-cycles, tiling-cycles and a new kind of path on the possible largest trapezoid grid which are needed for the following sections. We describe the basic rule of the transformability of the paths and the cycles into each other and extend our grids to larger graphs. In Sections 2 and 3 we define two kinds of graphs related to a checked fractal pattern on the generalized Sierpiński Gasket. We continue our observations with the basic properties of these triangular fractal approximating graphs independently of the recursive curves. We will describe the numbers of their vertices and edges, and their covering paths and cycles in general case with recursive and explicit formulas. Some of their cardinality specify new integer sequences. We also find the bijective relations between these formations.

Nonlinearity, 2007

Even though the open set condition (OSC) is generally accepted as the right condition to control overlaps of self-similar sets, it seems not clear how it relates to the actual size of the overlap. For connected self-similar sets in the plane, we prove that finite overlap implies OSC. On the other hand, there are Cantor sets with arbitrary small dimension which do not fulfil OSC.

Chaos, Solitons & Fractals

In this paper, we aim to construct fractal interpolation function(FIF) on the product of two Sierpiński gaskets. Further, we collect some results regarding smoothness of the constructed FIF. We prove, in particular, that the FIF are Hölder functions under specific conditions. In the final section, we obtain some bounds on the fractal dimension of FIF.

Fractals, 2014

Given a multi-valued function Φ : X ⊸ X on a topological space X we study the properties of its fixed fractal[Formula: see text], which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals [Formula: see text] and [Formula: see text] for a contracting compact-valued function Φ : X ⊸ X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.

Convex and Fractal Geometry, 2009

In the class of self-affine sets on R n we study a subclass for which the geometry is rather tractable. A type is a standardized position of two intersecting pieces. For a self-affine tiling, this can be identified with an edge or vertex type. We assume that the number of types is finite. We study the topology of such fractals and their boundary sets, and we show how new finite type fractals can be constructed. For finite type self-affine tiles in the plane we give an algorithm which decides whether the tile is homeomorphic to a disk.

Mathematische Nachrichten, 2007

We suggest a flexible way to study the self-similar interface of two different fractals. In contrast to previous methods the participating energies are modified in the neighborhood of the intersection of the fractals. In the example of the Vicsek snowflake and the 3-gasket, a variant of the Sierpinski gasket, we calculate the admissible transition constants via the "Short-cut Test". The resulting range of values is reinterpreted in terms of traces of Lipschitz spaces on the intersection. This allows us to describe the interface effects of different transition constants and indicates the techniques necessary to generalize the present interface results.