Post-critically finite fractal and Martin boundary

Authors:
Hongbing Ju, Ka-Sing Lau and Xiang-Yang Wang

Journal:
Trans. Amer. Math. Soc. **364** (2012), 103-118

MSC (2010):
Primary 28A78; Secondary 28A80

Published electronically:
August 4, 2011

MathSciNet review:
2833578

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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.

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Additional Information

**Hongbing Ju**

Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

**Ka-Sing Lau**

Affiliation:
Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Email:
kslau@math.cuhk.edu.hk

**Xiang-Yang Wang**

Affiliation:
School of Mathematics and Computational Science, Sun Yat-Sen University, Guang-Zhou 510275, People’s Republic of China

Email:
mcswxy@mail.sysu.edu.cn

DOI:
http://dx.doi.org/10.1090/S0002-9947-2011-05270-0

Keywords:
Fractals,
Green function,
harmonic functions,
monocyclic,
post critically finite,
Martin boundary,
transition probability.

Received by editor(s):
October 12, 2009

Received by editor(s) in revised form:
December 4, 2009

Published electronically:
August 4, 2011

Additional Notes:
This research was partially supported by a HKRGC grant and a Direct Grant from CUHK

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.