Golden Gaskets: Variations on the Sierpiński Sieve

2018

In this paper we have done some investigation on some concepts of the theory of metric space to analyze fractal objects. If we iteratively apply a finite set of contraction mappings to any point on a compact metric space, we will come arbitrarily close to a set of points in the space which is very often fractal. The present work addresses the problem of how iterated function systems may be used to construct such fractal objects. For this purpose, we discuss two algorithms producing fractals, namely that of deterministic algorithm, and random iterated algorithm. We have also discussed about the connection between Hausdorff dimension and iterated function systems.

2013

Chapter 6. Definition of Apollonian gasket 6.1. Basic facts Info G. Fibonacci numbers 6.2. Examples of non-bounded Apollonian tiling 6.3. Two interpretations of the set D 6.4. Generalized Descartes theorem 6.5. Integral solutions to Descartes equation Info H. Structure of some groups generated by reflections Chapter 7. Arithmetic properties of Apollonian gaskets 7.1. The structure of Q 7.2. Rational parametrization of circles 7.3. Nice parametrizations of discs tangent to a given disc 7.4. Integral Apollonian gaskets Info I. Möbius inversion formula Chapter 8. Geometric and group-theoretic approach Info J. Hyperbolic (Lobachevsky) plane L 8.1. Action of the group G and Apollonian gaskets 8.2. Action of the group Γ 4 on a Apollonian gasket Chapter 9. Many-dimensional Apollonian gaskets 9.1. General approach 9.2. 3-dimensional Apollonian gasket Bibliography A. Popular books, lectures and surveys B. Books C. Research papers D. Web sites

Journal of Mathematical Analysis and Applications, 1992

131) have considered a class of real functions whose graphs are, in general, fractal sets in R2. In this paper we give sufficient conditions for the fractal and Hausdorff dimensions to be equal for a certain subclass of fractal functions. The sets we consider are examples of self-affine fractals generated using iterated function systems (i.f.s.). Falconer [S] has shown that for almost all such sets the fractal and Hausdorff dimensions are equal and he gives a formula for the common dimension, due originally to Moran [S]. These results, however, give no information about individual fractal functions, In this paper we extend Moran's original method and show that if certain conditions on the i.f.s. are satisfied, then the two dimensions are equal. Kono [ 111 and Bedford [ 121 considered special cases of the subclass of fractal functions that we will introduce. Bedford and Urbanski [13] use a nonlinear setting to present conditions for the equality of Hausdorff and fractal dimension. However, their criteria are based on measure-theoretic characterizations and the use of the concept of generalized pressure. Our criterion on the other hand is based on the underlying geometry of the attractor and is easier to verify. We will show this on two specific examples which are more general than the self-ahine functions presented in [13].

Mathematische Nachrichten, 2016

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems Φ of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non-faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary of our main results, we obtain the non-faithfullness of the family of cylinders generated by the classical Lüroth expansion. Possible infinite entropy of the stochastic vector Q∞ which determines the metric relations for partitions of the generalized Lüroth expansions, possible non-faithfulness of the family Φ(Q∞) and the absence of general formulae for the calculation of the Hausdorff dimension even for probability measures with independent identically distributed symbols of generalized Lüroth expansions are all facts that do not allow to apply usual probabilistic methods, as well as methods of dynamical systems to study fractal properties of corresponding subsets of non-normal numbers. Since the 'divergent points technique' is not developed for measures generated by infinite IFS, the corresponding methods are also not applicable to solve the problem of the above fractal properties. Despite of the above mentioned difficulties, we develop new approach to the study of subsets of Q∞essentially non-normal numbers and prove (without any additional restrictions on the stochastic vector Q∞) that this set is superfractal. This result answers the open problem mentioned in [2] and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.

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.

2004

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as 'similarity' maps. Selfsimilar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of 'similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel's theorem on finiteness of integral points on hyperbolic curves and a fractal version of Falting's theorem on Diophantine approximation on abelian varieties.

Proceedings of the American Mathematical Society, 1992

For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express "absence of overlap" in terms of discontinuous action of a family of similitudes, thus improving the usual "open set condition".

Probability Theory and Related Fields, 1987

In a previous paper [21], the authors obtained tube formulas for certain fractals under rather general conditions. Based on these formulas, we give here a characterization of Minkowski measurability of a certain class of self-similar tilings and self-similar sets. Under appropriate hypotheses, self-similar tilings with simple generators (more precisely, monophase generators) are shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type. Under a natural geometric condition on the tiling, the result is transferred to the associated self-similar set (i.e., the fractal itself). Also, the latter is shown to be Minkowski measurable if and only if the associated scaling zeta function is of nonlattice type.

Bulletin of Mathematical Sciences, 2013

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.