Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

Distances in Sierpiński graphs and on the Sierpiński gasket

The well known planar fractal called the Sierpiński gasket can be defined with the help of a related sequence of graphs {G _n } _{n ≥ 0}, where G _n is the n-th Sierpiński graph, embedded in the Euclidean plane. In the present paper we prove geometric criteria that allow us to decide, whether a shortest path between two distinct vertices x and y in G _n , that lie in two neighbouring elementary triangles (of the same level), goes through the common vertex of the triangles or through two distinct vertices (both distinct from the common vertex) of those triangles. We also show criteria for the analogous problem on the planar Sierpiński gasket and in the 3-dimensional Euclidean space.     

Coloring Hanoi and Sierpiński graphs

It is shown that all Hanoi and Sierpiński graphs are in edge- and total coloring class 1, except those isomorphic to a complete graph of odd or even order, respectively. New proofs for their classification with respect to planarity are also given.     

On two theorems of Sierpiński

A theorem of Sierpiński says that every infinite set Q of reals contains an infinite number of disjoint subsets whose outer Lebesgue measure is the same as that of Q. He also has a similar theorem involving Baire property. We give a general theorem of this type and its corollaries, strengthening classical results.     

Resistance in Higher-Dimensional Sierpiński Carpets

We study the effective resistance between disjoint compact sets relative to the n-th level approximation F _n to the generalized Sierpiski carpet in d dimensions. This yields a simple criterion for determining recurrence of simple random walk on the associated pre-fractal graph in terms of the resistance scaling factor.     

Sierpiński图和Sierpiński金字塔上的一些费马型指标（英文）

本文中我们给出一些费马型指标的结果,包括费马偏心距、费马半径和费马直径.我们使用编码的组合方法确定了Sierpiński图和Sierpiński金字塔的费马半径和费马直径.通过归一化Sierpiński图上的距离,我们给出了Sierpiński金字塔的平均费马偏心距的精确值并由此得到关于Sierpiński图的渐近公式.     

The Average Eccentricity of Sierpiński Graphs

We determine the eccentricity of an arbitrary vertex, the average eccentricity and its standard deviation for all Sierpiński graphs ${S_p^n}$ . Special cases are the graphs ${S_2^{n}}$ , which are isomorphic to the state graphs of the Chinese Rings puzzle with n rings and the graphs ${S_3^{n}}$ isomorphic to the Hanoi graphs ${H_3^{n}}$ representing the Tower of Hanoi puzzle with 3 pegs and n discs.     

On dynamics of the Sierpiński carpet

We prove that the Sierpiński curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.     

The average distance on the Sierpiński gasket

Summary The canonical distance of points on the Sierpiski gasket is considered and its expectation deduced. The solution is surprising, both for the value and for the method derived from an analysis of graphs connected with the Tower of Hanoi problem.     

A geometric property of the Sierpiński carpet

The main aim of this paper is to find formulae for the computation of the geodesic metric on the Sierpí nski carpet. This is accomplished by introducing carpet coordinates. Subsequently we show the equivalence of the Euclidean and the geodesic metric on this fractal.     

On a problem of Sierpiński,Ⅱ

For any integer s≥ 2, let μsbe the least integer so that every integer l μs is the sum of exactly s integers which are pairwise relatively prime. In 1964, Sierpi′nski asked for the determination of μs. Let pibe the i-th prime and let μs= p2 + p3 + + ps+1+ cs. Recently, the authors solved this problem. In particular,we have(1) cs=-2 if and only if s = 2;(2) the set of integers s with cs= 1100 has asymptotic density one;(3) cs∈ A for all s ≥ 3, where A is an explicit set with A ■[2, 1100] and |A| = 125. In this paper, we prove that,(1) for every a ∈ A, there exists an index s with cs= a;(2) under Dickson's conjecture, for every a ∈ A,there are infinitely many s with cs= a. We also point out that recent progress on small gaps between primes can be applied to this problem.     

A topological characterization of the Sierpiński triangle

We present a topological characterization of the Sierpiński triangle. This answers question 58 from the Problem book of the Open Problem Seminar held at Charles University. In fact we give a characterization of the Apollonian gasket first. Consequently we show that any subcontinuum of the Apollonian gasket, whose boundary consists of three points, is homeomorphic to the Sierpiński triangle.     

Szegö Limit Theorems on the Sierpiński Gasket

We use the existence of localized eigenfunctions of the Laplacian on the Sierpiński gasket (SG) to formulate and prove analogues of the strong Szegö limit theorem in this fractal setting. Furthermore, we recast some of our results in terms of equally distributed sequences.     

Equidistribution and Brownian motion on the Sierpiński gasket

We introduce several concepts of discrepancy for sequences on the Sierpiski gasket. Furthermore a law of iterated logarithm for the discrepancy of trajectories of Brownian motion is proved. The main tools for this result are regularity properties of the heat kernel on the Sierpiski gasket. Some of the results can be generalized to arbitrary nested fractals in the sense of T. Lindstrøm.With 2 FiguresDedicated to Prof. Edmund Hlawka on the occasion of his 80^th birthdayThe authors are supported by the Austrian Science Foundation project Nr. P10223-PHY and by the Austrian-Italian scientific cooperation program project Nr. 39     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Sierpiński graphs as spanning subgraphs of Hanoi graphs

Hanoi graphs H _p ⁿ model the Tower of Hanoi game with p pegs and n discs. Sierpinski graphs S _p ⁿ arose in investigations of universal topological spaces and have meanwhile been studied extensively. It is proved that S _p ⁿ embeds as a spanning subgraph into H _p ⁿ if and only if p is odd or, trivially, if n = 1.