Maximal inequality and ergodic theorems for Markov groups

A. I. Bufetov ; A. V. Klimenko

Informatics and Automation, Mathematical control theory and differential equations, Tome 277 (2012), pp. 33-48.

Résumé

The paper shows that for actions of Markov semigroups, in particular, of finitely generated word hyperbolic groups, the Cesàro means of spherical averages converge almost everywhere for any function from the class $L^p$, $p>1$.

Bibliographie

[1] Bowen L., “Invariant measures on the space of horofunctions of a word hyperbolic group”, Ergodic Theory Dyn. Syst., 30:1 (2010), 97–129 | DOI | MR | Zbl

[2] Bowen L., Nevo A., Geometric covering arguments and ergodic theorems for free groups, E-print, 2009, arXiv: 0912.4953v2 [math.DS] | MR

[3] Bufetov A.I., “Ergodicheskie teoremy dlya deistvii neskolkikh otobrazhenii”, UMN, 54:4 (1999), 159–160 | DOI | MR | Zbl

[4] Bufetov A.I., “Operatornye ergodicheskie teoremy dlya deistvii svobodnykh polugrupp i grupp”, Funkts. analiz i ego pril., 34:4 (2000), 1–17 | DOI | MR | Zbl

[5] Bufetov A.I., “Markov averaging and ergodic theorems for several operators”, Topology, ergodic theory, real algebraic geometry: Rokhlin's memorial, AMS Transl. Ser. 2, 202, Amer. Math. Soc., Providence, RI, 2001, 39–50 | MR | Zbl

[6] Bufetov A.I., “Convergence of spherical averages for actions of free groups”, Ann. Math. Ser. 2, 155 (2002), 929–944 | DOI | MR | Zbl

[7] Bufetov A.I., Khristoforov M., Klimenko A., “Cesàro convergence of spherical averages for measure-preserving actions of Markov semigroups and groups”, Int. Math. Res. Not., 2011 | DOI | MR

[8] Bufetov A.I., Series C., “A pointwise ergodic theorem for Fuchsian groups”, Math. Proc. Cambridge Philos. Soc., 151 (2011), 145–159 | DOI | MR | Zbl

[9] Calegari D., Fujiwara K., “Combable functions, quasimorphisms, and the central limit theorem”, Ergodic Theory Dyn. Syst., 30:5 (2010), 1343–1369 | DOI | MR | Zbl

[10] Cannon J.W., “The combinatorial structure of cocompact discrete hyperbolic groups”, Geom. Dedicata, 16:2 (1984), 123–148 | DOI | MR | Zbl

[11] Dunford N., Schwartz J.T., Linear operators. Part I: General theory, Applied Mathematics, 7, Interscience, New York, 1958 | MR | Zbl

[12] Fujiwara K., Nevo A., “Maximal and pointwise ergodic theorems for word-hyperbolic groups”, Ergodic Theory Dyn. Syst., 18:4 (1998), 843–858 | DOI | MR | Zbl

[13] Sur les groupes hyperboliques d'après Mikhael Gromov, Prog. Math., 83, Ed. by É. Ghys, P. de la Harpe, Birkhäuser, Boston, 1990 | MR | Zbl

[14] Grigorchuk R.I., “Individualnaya ergodicheskaya teorema dlya deistvii svobodnykh grupp”, Tez. dokl. XII Shkoly po teorii operatorov v funktsionalnykh prostranstvakh, Ch. 1, Tambov, 1987, 57

[15] Grigorchuk R.I., “Ergodicheskie teoremy dlya deistvii svobodnoi gruppy i svobodnoi polugruppy”, Mat. zametki, 65:5 (1999), 779–783 | DOI | MR | Zbl

[16] Grigorchuk R.I., “Ergodicheskaya teorema dlya deistvii svobodnoi polugruppy”, Tr. MIAN, 231, 2000, 119–133 | MR | Zbl

[17] Gromov M., “Hyperbolic groups”, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, Springer, New York, 1987, 75–263 | DOI | MR

[18] Nevo A., “Harmonic analysis and pointwise ergodic theorems for noncommuting transformations”, J. Amer. Math. Soc., 7:4 (1994), 875–902 | DOI | MR | Zbl

[19] Nevo A., “Pointwise ergodic theorems for actions of groups”, Handbook of dynamical systems, V. 1B, Elsevier, Amsterdam, 2006, 871–982 | DOI | MR | Zbl

[20] Nevo A., Stein E.M., “A generalization of Birkhoff's pointwise ergodic theorem”, Acta math., 173 (1994), 135–154 | DOI | MR | Zbl

[21] Pollicott M., Sharp R., “Ergodic theorems for actions of hyperbolic groups”, Proc. Amer. Math. Soc. (to appear) | MR