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.     

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.     

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.     

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.     

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.     

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.     

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     

Harmonic analysis on the Sierpiński gasket and singular functions

We study the restriction on [0,1] of harmonic functions on the Sierpiński gasket, proving they are singular functions whenever they are monotone. We show that their derivatives are zero or infinity on certain non-denumerable sets. Finally, we show they are among a wider class of functions that contains some already known and studied functions.     

Generalized Power Domination: Propagation Radius and Sierpiński Graphs

The recently introduced concept of k-power domination generalizes domination and power domination, the latter concept being used for monitoring an electric power system. The k-power domination problem is to determine a minimum size vertex subset S of a graph G such that after setting X=N[S], and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of v are not yet in X, we get X=V(G). In this paper the k-power domination number of Sierpiński graphs is determined. The propagation radius is introduced as a measure of the efficiency of power dominating sets. The propagation radius of Sierpiński graphs is obtained in most of the cases.     

Equivalent Semi-Norms of Non-Local Dirichlet Forms on the Sierpiński Gasket and Applications

We construct equivalent semi-norms of non-local Dirichlet forms on the Sierpiński gasket and apply these semi-norms to a convergence problem and a trace problem. We also construct explicitly a sequence of non-local Dirichlet forms with jumping kernels equivalent to |x ? y|^?α?β that converges exactly to local Dirichlet form.     

Vertex-, edge-, and total-colorings of Sierpiński-like graphs

Vertex-colorings, edge-colorings and total-colorings of the Sierpiński gasket graphs S_n, the Sierpiński graphs S(n,k), graphs S⁺(n,k), and graphs S⁺⁺(n,k) are considered. In particular, χ^″(S_n), χ^′(S(n,k)), χ(S⁺(n,k)), χ(S⁺⁺(n,k)), χ^′(S⁺(n,k)), and χ^′(S⁺⁺(n,k)) are determined.     

Multiple solutions for a class of nonlinear elliptic equations on the Sierpiński gasket

This paper investigates a class of nonlinear elliptic equations on a fractal domain. We establish a strong Sobolev-type inequality which leads to the existence of multiple non-trivial solutions of Δu +c(x)u =f(x,u), with zero Dirichlet boundary conditions on the Sierpiski gasket. Our existence results do not require any growth conditions off(x, t) in t, in contrast to the classical theory of elliptic equations on smooth domains.     

The generating functions of the rank and crank modulo 8

Let N(i,m;n) be the number of partitions of n with rank (Dyson) congruent to i (mod m) and let M(j,m;n) be the number of partitions of n with crank (Andrews, Garvan) congruent to j (mod m). I give here the generating functions for the numbers N(i,8;n) and M(j,8;n). I suggest forms for the one hundred power series

Lifschitz singularity for subordinate Brownian motions in presence of the Poissonian potential on the Sierpiński gasket

We establish the Lifschitz-type singularity around the bottom of the spectrum for the integrated density of states for a class of subordinate Brownian motions in presence of the nonnegative Poissonian random potentials, possibly of infinite range, on the Sierpiński gasket. We also study the long-time behaviour for the corresponding averaged Feynman–Kac functionals.     

Sierpiński Gasket as a Martin Boundary I: Martin Kernels; Dedicated to Professor Masatoshi Fukushima on the occasion of his 60th birthday

We show that a Sierpiski gasket in N dimension is homeomorphic to the minimal Martin boundary of some canonical Markov chain. This provides a new class of examples for the boundary theory of Markov chains and the basis for a harmonic analysis on p.c.f. fractal structures.