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 Full-text PDF
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
[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
(39 #6408)
[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
(2005h:28022)
[W]Wolfgang
Woess, Random walks on infinite graphs and groups, Cambridge
Tracts in Mathematics, vol. 138, Cambridge University Press,
Cambridge, 2000. MR 1743100
(2001k:60006)
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
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)
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)
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)
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