Reflections on Harmonic Analysis of the Sierpiński Gasket

Denker, Manfred; Satô, Hiroshi

doi:10.1002/1522-2616(200207)241:1<32::aid-mana32>3.0.co;2-5

Cited by 14 publications

(8 citation statements)

References 7 publications

(21 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In fact, S is bi-Lipschitz equivalent to the quotient A N / ∼ . As shown by the authors in [4], every S--harmonic function φ on S is determined uniquely by the function values I(φ)(w), w ∈ A − , moreover I(φ) is A -harmonic and diagonal (i.e. I(φ)(w) = I(φ)(wτ (w)) for all w ∈ A − ).…”

Section: Applicationmentioning

confidence: 97%

“…7 we collect some well known facts about iterated function systems and word spaces. Moreover, we review some results by Denker & Sato on the representation of the Sierpiński gasket as a Martin boundary from [2][3][4]. Finally we sketch how to apply the Furstenberg-type formulas from Sec.…”

Section: Introductionmentioning

confidence: 99%

“…Dynkin [6]). Using this fact Denker and his co-workers developed a probabilistic approach to a (harmonic) analysis on the Sierpiński gasket in [2][3][4][5]. The latter is the best known example of a p.c.f.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Furstenberg-Type Formulas Over Shift Spaces

Koch

2003

Stoch. Dyn.

View full text Add to dashboard Cite

We define a class of Markov chains of unbounded range on word spaces and deduce a Furstenberg-type integral representation for a subspace of the P -harmonic functions. As an application we obtain a Furstenberg-type formula for a set of continuous harmonic functions on p.c.f. self-similar sets.Closely connected to the notion of P -harmonic functions are different boundaries of the Markov chain P . On the one hand, there is the topological notion of the Martin boundary. This is the set of possible limits of the chain at infinity, it arises from a compactification of the state space X and is equipped with a collection of hitting distributions (being invariant under the adjoint of the Markov operator P and therefore harmonic). This allows an integral representation of all P -harmonic functions in terms of the potential theoretic Martin kernel (see e.g. Dynkin [6]). On the other hand, there is the notion of the Poisson boundary of a Markov chain. Different interpretations can be found in the literature (see e.g. Furstenberg [8], 499 Stoch. Dyn. 2003.03:499-527. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ DAVIS on 02/08/15. For personal use only.

show abstract

Section: Applicationmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Furstenberg-Type Formulas Over Shift Spaces

Koch

2003

Stoch. Dyn.

View full text Add to dashboard Cite

show abstract

“…Denker and Sato [5,6] created a Markov chain whose Martin boundary is homeomorphic to the Sierpiński gasket (see Figure 1), and used potential theory on the Martin boundary to induce a harmonic structure. In [7] they identified a subclass of 'strongly harmonic functions' on the Martin boundary which coincides with Kigami's canonical class of harmonic functions [16,17,29]. Denker, Imai and Koch [4] extended this construction to some non-self-similar Sierpiński type gaskets and studied an associated Dirichlet form.…”

Section: Introductionmentioning

confidence: 99%

The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

2020

View full text Add to dashboard Cite

Link to publication on Research at Birmingham portal General rights Unless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or the copyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposes permitted by law. • Users may freely distribute the URL that is used to identify this publication. • Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of private study or non-commercial research. • User may use extracts from the document in line with the concept of 'fair dealing' under the Copyright, Designs and Patents Act 1988 (?) • Users may not further distribute the material nor use it for the purposes of commercial gain. Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document. When citing, please reference the published version. Take down policy While the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.

show abstract

“…For a contractive iterated function system (IFS) {S j } N j=1 with an invariant set K, there is a canonical tree structure on the symbolic space that represents K. Recently, Denker and Sato [DS1,DS2] proved that for the special case of Sierpiński gasket, there is a nature transition probability on the tree so that the Martin boundary of the associated Markov chain is homeomorphic to K. Moreover, in [DS3] they identified a subclass of "strongly harmonic functions" on the Sierpiński gasket that coincides with Kigami's harmonic functions [K1, K2]. The case of the pentagasket and other extensions were studied in [I] and [DIK].…”

Section: Introductionmentioning

confidence: 99%

Post-critically finite fractal and Martin boundary

Lau

Wang

2011

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For an iterated function system (IFS), which we call simple post critically finite, we introduce a Markov chain on the corresponding symbolic space and study its boundary behavior. We carry out some fine estimates of the Martin metric and use them to prove that the Martin boundary can be identified with the invariant set (fractal) of the IFS. This enables us to bring in the boundary theory of Markov chains and the discrete potential theory on this class of fractal sets.

show abstract

Reflections on Harmonic Analysis of the Sierpiński Gasket

Cited by 14 publications

References 7 publications

Furstenberg-Type Formulas Over Shift Spaces

Furstenberg-Type Formulas Over Shift Spaces

The Sierpiński gasket as the Martin boundary of a non-isotropic Markov chain

Post-critically finite fractal and Martin boundary

Contact Info

Product

Resources

About