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

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

Published electronically:
August 4, 2011

MathSciNet review:
2833578

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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]**A. Ancona,*Positive harmonic functions and hyperbolicity*, Potential Theory. Surveys and Problems, (eds, J. Kral,*et al.*), Lecture Notes in Math., no. 1344, Springer, 1987, 128-136. MR**973878****[BR]**C. Bandt and H. Rao,*Topology and separation of self-similar fractals in the plane*, Nonlinearity,**20**(2007), 1463-1474. MR**2327133 (2008g:28036)****[D]**J. Doob,*Discrete potential theory and boundaries*, J. Math. and Mech.,**8**(1959), 433-458. MR**0107098 (21:5825)****[DIK]**M. Denker, A. Imai and S. Koch,*Dirichlet forms on quotients of shift spaces*, Colloq. Math.,**107**(2007), 57-80. MR**2283132 (2008b:28010)****[DL]**Q. Deng and K. S. Lau,*Open set condition and post-critically finite self-similar sets*, Nonlinearity,**21**(2008), 1227-1232. MR**2422376 (2009j:28020)****[DS1]**M. Denker and H. Sato,*Sierpiński gasket as a Martin boundary I: Martin kernel*, Potential Anal.,**14**(2001), 211-232. MR**1822915 (2002f:60139)****[DS2]**M. Denker and H. Sato,*Sierpiński gasket as a Martin boundary II: The intrinsic metric*, Publ. RIMS, Kyoto Univ.,**35**(1999), 769-794. MR**1739300 (2002f:60140)****[DS3]**M. Denker and H. Sato,*Reflections on harmonic analysis of the Sierpiński gasket*, Math. Nachr.**241**(2002), 32-55. MR**1912376 (2003e:28016)****[Dy]**E. Dynkin,*Boundary theory of Markov processes (the discrete case)*, Russian Math. Surveys,**24**(1969), 1-42. MR**0245096 (39:6408)****[F]**K. Falconer,*Fractal geometry*, Mathematical Foundations and Applications, Wiley, 1990. MR**1102677 (92j:28008)****[H]**G. Hunt,*Markov chains and Martin boundaries*, Illinois J. Math.,**4**(1960), 313-340. MR**0123364 (23:A691)****[Hu]**J. Hutchinson,*Fractal and self-similarity*, Indiana Univ. Math. J.,**30**(1981), 712-747. MR**625600 (82h:49026)****[I]**A. Imai,*The difference between letters and a Martin kernel of a modulo Markov chain*, Adv. in Appl. Math.**28**(2002), no. 1, 82-106. MR**1884389 (2003f:60126)****[K1]**J. Kigami,*Analysis on Fractals*, Cambridge University Press, 2001. MR**1840042 (2002c:28015)****[K2]**J. Kigami,*Harmonic calculus on p.c.f. self-similar sets*, Trans. Amer. Math. Soc.,**335**(1993), 721-755. MR**1076617 (93d:39008)****[Ka]**V. Kaimanovich,*Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization*, Fractals in Graz 2001, 145-183, Trends Math., Birkhäuser, 2003. MR**2091703 (2005h:28022)****[LW]**K. S. Lau and X. Y. Wang,*Self-similar sets as hyperbolic boundary*, Indiana Univ. Math. J.,**58**(2009), 1777-1795. MR**2542979****[M]**R. Martin,*Minimal positive harmonic functions*, Trans. Amer. Math. Soc.,**49**(1941), 137-172. MR**0003919 (2:292h)****[P]**R. Phelps, Lectures on Choquet's theorem, 2rd ed., Lectures Notes in Math., no. 1757, Springer, 2001. MR**1835574 (2002k:46001)****[S1]**R. Strichartz,*Differential equations on fractals: a tutorial*, Princeton University Press, 2006. MR**2246975 (2007f:35003)****[S2]**R. Strichartz,*The Laplacian on the Sierpiński gasket via the method of averages*, Pacific J. Math.,**201**(2001), 241-256. MR**1867899 (2003f:35056)****[S3]**R. Strichartz,*Analysis on product of fractals*, Trans. Amer. Math. Soc.,**355**(2003), 4019-4043. MR**1990573 (2004b:28013)****[W]**W. Woess,*Random walks on infinite graphs and groups*, Cambridge University Press, 2000. MR**1743100 (2001k:60006)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2010):
28A78,
28A80

Retrieve articles in all journals with MSC (2010): 28A78, 28A80

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.