Martin Boundary and Exit Space on the Sierpinski Gasket

Ka Sing Lau; Sze Man Ngai

doi:10.1007/s11425-011-4339-x

Martin Boundary and Exit Space on the Sierpinski Gasket

Ka Sing Lau, Sze Man Ngai

Mathematical Sciences

The Chinese University of Hong Kong

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

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.

Original language	American English
Journal	Science China Mathematics
Volume	55
DOIs	https://doi.org/10.1007/s11425-011-4339-x
State	Published - Jan 1 2012

Disciplines

Education
Mathematics

Keywords

28A80
31C35
60J10
60J50
Fractal
Green function
Harmonic function
Markov chain
Martin boundary
Sierpinski gasket

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http://dx.doi.org/10.1007/s11425-011-4339-x

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title = "Martin Boundary and Exit Space on the Sierpinski Gasket",

abstract = "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.",

keywords = "28A80, 31C35, 60J10, 60J50, Fractal, Green function, Harmonic function, Markov chain, Martin boundary, Sierpinski gasket",

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AU - Ngai, Sze Man

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N2 - 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.

AB - 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.

KW - 28A80

KW - 31C35

KW - 60J10

KW - 60J50

KW - Fractal

KW - Green function

KW - Harmonic function

KW - Markov chain

KW - Martin boundary

KW - Sierpinski gasket

UR - https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/127

UR - http://dx.doi.org/10.1007/s11425-011-4339-x

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DO - 10.1007/s11425-011-4339-x

M3 - Article

SN - 1674-7283

VL - 55

JO - Science China Mathematics

JF - Science China Mathematics

ER -

Martin Boundary and Exit Space on the Sierpinski Gasket

Abstract

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Keywords

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