Abstract
New ergodic theorems are obtained for measure-preserving actions of free semigroups and groups. These theorems are derived from ergodic theorems for Markov operators. This approach also allows one to obtain ergodic theorems for some classes of Markov semigroups. Results of the paper generalize classical ergodic theorems of Kakutani, Oseledets, and Guivarc'h, and recent ergodic theorems of Grigorchuk, Nevo, and Nevo and Stein.
Similar content being viewed by others
References
M. Gromov, “Hyperbolic groups,” In: Essays in Group Theory (Gersten S. M., ed.), M.S.R.I. Publ., Vol. 8, Springer-Verlag, 1987, pp. 75-263.
V. I. Oseledets, “Markovc hains, skew-products and ergodic theorems for ‘general’ dynamical systems,” Teor. Veroyatn. Primenen., 10, No. 3, 551-557 (1965).
S. Kakutani, “Random ergodic theorems and Markoff processes with a stable distribution,” In: Proc. 2nd Berkeley Symposium Math. Stat. and Prob., 1951, pp. 247-261.
A. M. Vershik, “Dynamical theory of growth in groups: entropy, boundaries, examples,” Usp. Mat. Nauk, 55, No. 4, 59-128 (2000).
A. M. Vershik, S. Nechaev, and R. Bikbov, Statistical Properties of Braid Groups in Locally Free Approximation, Preprint IHES/M/99/45, June 1999.
A. M. Vershik, “Numerical characteristics of groups and the relations between them,” Zap. Nauchn. Sem. POMI, 256, 5-18 (1999).
R. I. Grigorchuk, “Individual ergodic theorem for actions of free groups,” In: Proceedings of the Tambov Workshop in the Theory of Functions, 1986. 251
R. I. Grigorchuk, “Ergodic theorems for actions of a free group and a free semigroup,” Mat. Zametki, 65, No. 5, 779-782 (1999).
V. I. Arnold and A. L. Krylov, “Uniform distribution of points on a sphere and certain ergodic properties of solutions of linear ordinary differential equations in a complex domain,” Dokl. Akad. Nauk SSSR, 148, No. 1, 9-12 (1963).
L. A. Grigorenko, “σ-algebra of symmetric events for a countable Markov chain,” Teor. Veroyatn. Primenen., 24, No. 1, 198-204 (1979).
Z. I. Bezhaeva and V. I. Oseledets, “On the symmetric σ-algebra of a stationary Harris Markovpro cess,” Teor. Veroyatn. Primenen., 41, No. 4, 869-877 (1996).
N. Dunford and J. T. Schwartz, “Convergence almost everywhere of operator averages,” J. Rational Mech. Anal., 5, 129-178 (1956).
E. Hopf, “The general temporally discrete Markoff process,” J. Rational Mech. Anal., 3, 13-45 (1954).
A. N. Kolmogorov, “A local limit theorem for classical Markov chains,” Izv. Akad. Nauk SSSR, Ser. Mat., 13, No. 4, 281-300 (1949); English transl. in Select. Transl. Math. Statist. Probab., Vol. 2, Amer. Math. Soc., 1962, pp. 109-129.
B. M. Gurevich, “The local limit theorem for Markov chains and regularity type conditions,” Teor. Veroyatn. Primenen., 13, No. 1, 183-190 (1968).
A. Nevo, “Harmonic analysis and pointwise ergodic theorems for non-commuting transformations,” J. Amer. Math. Soc., 7, 875-902 (1994).
J. Cannon, “The combinatorial structure of cocompact discrete hyperbolic groups,” Geom. Dedicata, 16, 123-148 (1984).
A. Nevo and E. M. Stein, “A generalization of Birkhoff's pointwise ergodic theorem,” Acta Math., 173, 135-154 (1994).
Y. Guivarc'h, “Généralisation d'un théorème de von Neumann,” C. R. Acad. Sci Paris, 268, 1020-1023 (1969).
E. R. Lorch, “Means of iterated transformations in reflexive vector spaces,” Bull. Amer. Math. Soc., 45, 945-947 (1939).
U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin-New York, 1985.
A. Bufetov, “Ergodic theorems for several maps,” Usp. Mat. Nauk, 54, No. 4, 159-160 (1999).
M. Akcoglu, “A pointwise ergodic theorem in Lp-spaces,” Canad. J. Math., 27, 1075-1082 (1975).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bufetov, A.I. Operator Ergodic Theorems for Actions of Free Semigroups and Groups. Functional Analysis and Its Applications 34, 239–251 (2000). https://doi.org/10.1023/A:1004116205980
Issue Date:
DOI: https://doi.org/10.1023/A:1004116205980