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.     

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.     

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.     

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.     

Martin boundary and exit space on the Sierpinski gasket

We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.     

Hausdorff dimension of the limit sets of some planar geometric constructions

We determine the Hausdorff and box dimension of the limit sets for some class of planar non-Moran-like geometric constructions generalizing the Bedford-McMullen general Sierpiński carpets. The class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines, as well as some non-affine examples, e.g. the flexed Sierpiński gasket.     

Harnack inequality for symmetric stable processes on fractalsL'inégalité de Harnack pour les processus symétriques stables sur les fractals

We study nonnegative harmonic functions of symmetric α-stable processes on d-sets F. We prove the Harnack inequality for such functions when α∈(0,2/d_w)∪(d_s,2). Furthermore, we investigate the decay rate of harmonic functions and the Carleson estimate near the boundary of a region in F. In the particular case of natural cells in the Sierpiński gasket we also prove the boundary Harnack principle. To cite this article: K. Bogdan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 59–63.     

Qualitative Analysis of Gradient-Type Systems with Oscillatory Nonlinearities on the Sierpi′nski Gasket

Under an appropriate oscillating behavior either at zero or at infinity of the nonlinear data, the existence of a sequence of weak solutions for parametric quasilinear systems of the gradient-type on the Sierpi′nski gasket is proved. Moreover, by adopting the same hypotheses on the potential and in presence of suitable small perturbations, the same conclusion is achieved. The approach is based on variational methods and on certain analytic and geometrical properties of the Sierpi′nski fractal as, for instance, a compact embedding result due to Fukushima and Shima.     

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.     

Analytic Functions in Local Shift-Invariant Spaces and Analytic Limits of Level Dependent Subdivision

Journal of Fourier Analysis and Applications - We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ( $$SG$$ ), which is a fractal set that can be viewed as a limit...     

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.     

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.     

On powers associated with Sierpiński numbers, Riesel numbers and Polignac's conjecture

We address conjectures of P. Erd?s and conjectures of Y.-G. Chen concerning the numbers in the title. We obtain a variety of related results, including a new smallest positive integer that is simultaneously a Sierpiński number and a Riesel number and a proof that for every positive integer r, there is an integer k such that the numbers k,k²,k³,…,kr are simultaneously Sierpiński numbers.     

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.     

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.     

Packing coloring of Sierpiński-type graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in \{1,\ldots ,k\}\), where each \(V_i\) is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs \(ST^n_3\) is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs \(S^n_G\) with respect to all connected graphs G of order 4.     

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.     

Martin Boundary of a Fractal Domain

A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpiski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpiski gasket itself.     

Polymorphism clones of homogeneous structures: generating sets,Sierpiński rank,cofinality and the Bergman property

In this paper, motivated by classical results by Sierpiński, Arnold and Kolmogorov, we derive sufficient conditions for polymorphism clones of homogeneous structures to have a generating set of bounded arity. We use our findings in order to describe a class of homogeneous structures whose polymorphism clones have a finite Sierpiński rank, uncountable cofinality, and the Bergman property.     

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.