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     

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.     

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.     

Saddle-point solutions to Dirichlet problems on the Sierpiński gasket

We provide sufficient conditions for the existence of saddle-point solutions to a system driven by the weak Laplacian on the Sierpiński gasket. We analyze also its stability by proving its continuous dependence on parameters.     

Displacement exponent for loop-erased random walk on the Sierpiński gasket

We prove that loop-erased random walks on the finite pre-Sierpiński gaskets can be extended to a loop-erased random walk on the infinite pre-Sierpiński gasket by using the ‘erasing-larger-loops-first’ method, and obtain the asymptotic behavior of the walk as the number of steps increases, in particular, the displacement exponent and a law of the iterated logarithm.     

New results on variants of covering codes in Sierpiński graphs

In this paper we study identifying codes, locating-dominating codes, and total-dominating codes in Sierpiński graphs. We compute the minimum size of such codes in Sierpiński graphs.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.     

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.     

Connected generalised Sierpiński carpets

Generalised Sierpiński carpets are planar sets that generalise the well-known Sierpiński carpet and are defined by means of sequences of patterns. We present necessary and sufficient conditions, under which generalised Sierpiński carpets are connected, with respect to Euclidean topology.     

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.