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.

**[A]**Alano Ancona,*Positive harmonic functions and hyperbolicity*, Potential theory—surveys and problems (Prague, 1987) Lecture Notes in Math., vol. 1344, Springer, Berlin, 1988, pp. 1–23. MR**973878**, 10.1007/BFb0103341**[BR]**Christoph Bandt and Hui Rao,*Topology and separation of self-similar fractals in the plane*, Nonlinearity**20**(2007), no. 6, 1463–1474. MR**2327133**, 10.1088/0951-7715/20/6/008**[D]**J. L. Doob,*Discrete potential theory and boundaries*, J. Math. Mech.**8**(1959), 433–458; erratum 993. MR**0107098****[DIK]**Manfred Denker, Atsushi Imai, and Susanne Koch,*Dirichlet forms on quotients of shift spaces*, Colloq. Math.**107**(2007), no. 1, 57–80. MR**2283132**, 10.4064/cm107-1-7**[DL]**Qi-Rong Deng and Ka-Sing Lau,*Open set condition and post-critically finite self-similar sets*, Nonlinearity**21**(2008), no. 6, 1227–1232. MR**2422376**, 10.1088/0951-7715/21/6/004**[DS1]**Manfred Denker and Hiroshi Sato,*Sierpiński gasket as a Martin boundary. I. Martin kernels*, Potential Anal.**14**(2001), no. 3, 211–232. MR**1822915**, 10.1023/A:1011232724842**[DS2]**Manfred Denker and Hiroshi Sato,*Sierpiński gasket as a Martin boundary. II. The intrinsic metric*, Publ. Res. Inst. Math. Sci.**35**(1999), no. 5, 769–794. MR**1739300**, 10.2977/prims/1195143423**[DS3]**Manfred Denker and Hiroshi Sato,*Reflections on harmonic analysis of the Sierpiński gasket*, Math. Nachr.**241**(2002), 32–55. MR**1912376**, 10.1002/1522-2616(200207)241:1<32::AID-MANA32>3.0.CO;2-5**[Dy]**E. B. Dynkin,*The boundary theory of Markov processes (discrete case)*, Uspehi Mat. Nauk**24**(1969), no. 2 (146), 3–42 (Russian). MR**0245096****[F]**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****[H]**G. A. Hunt,*Markoff chains and Martin boundaries*, Illinois J. Math.**4**(1960), 313–340. MR**0123364****[Hu]**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, 10.1512/iumj.1981.30.30055**[I]**Atsushi Imai,*The difference between letters and a Martin kernel of a modulo 5 Markov chain*, Adv. in Appl. Math.**28**(2002), no. 1, 82–106. MR**1884389**, 10.1006/aama.2001.0768**[K1]**Jun Kigami,*Analysis on fractals*, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR**1840042****[K2]**Jun Kigami,*Harmonic calculus on p.c.f. self-similar sets*, Trans. Amer. Math. Soc.**335**(1993), no. 2, 721–755. MR**1076617**, 10.1090/S0002-9947-1993-1076617-1**[Ka]**Vadim A. Kaimanovich,*Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization*, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, pp. 145–183. MR**2091703****[LW]**Ka-Sing Lau and Xiang-Yang Wang,*Self-similar sets as hyperbolic boundaries*, Indiana Univ. Math. J.**58**(2009), no. 4, 1777–1795. MR**2542979**, 10.1512/iumj.2009.58.3639**[M]**Robert S. Martin,*Minimal positive harmonic functions*, Trans. Amer. Math. Soc.**49**(1941), 137–172. MR**0003919**, 10.1090/S0002-9947-1941-0003919-6**[P]**Robert R. Phelps,*Lectures on Choquet’s theorem*, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer-Verlag, Berlin, 2001. MR**1835574****[S1]**Robert S. Strichartz,*Differential equations on fractals*, Princeton University Press, Princeton, NJ, 2006. A tutorial. MR**2246975****[S2]**Robert S. Strichartz,*The Laplacian on the Sierpinski gasket via the method of averages*, Pacific J. Math.**201**(2001), no. 1, 241–256. MR**1867899**, 10.2140/pjm.2001.201.241**[S3]**Robert S. Strichartz,*Fractafolds based on the Sierpiński gasket and their spectra*, Trans. Amer. Math. Soc.**355**(2003), no. 10, 4019–4043 (electronic). MR**1990573**, 10.1090/S0002-9947-03-03171-4**[W]**Wolfgang Woess,*Random walks on infinite graphs and groups*, Cambridge Tracts in Mathematics, vol. 138, Cambridge University Press, Cambridge, 2000. MR**1743100**

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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:
https://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.