The basic ergodic theorems, yet again

  • Jairo Bochi Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Chile.
Keywords: Maximal ergodic theorem, Birkhoff’s ergodic theorem, Rokhlin lemma, Kingman’s subadditive ergodic theorem


A generalization of Rokhlin’s Tower Lemma is presented. The Maximal Ergodic Theorem is then obtained as a corollary. We also use the generalized Rokhlin lemma, this time combined with a subadditive version of Kac’s formula, to deduce a subadditive version of the Maximal Ergodic Theorem due to Silva and Thieullen.

In both the additive and subadditive cases, these maximal theorems immediately imply that “heavy” points have positive probability. We use heaviness to prove the pointwise ergodic theorems of Birkhoff and Kingman.


Avila, A.; Bochi, J.–On the subadditive ergodic theorem. Manuscript, ̃jairo.bochi/docs/kingbirk.pdf

Birkhoff, G.D.– Proof of the ergodic theorem.Proc. Nat. Acad. Sci. USA, 17 (1931),656–660.

Heinemann, S.-M.; Schmitt, O.– Rokhlin’s lemma for non-invertible maps. Dynam. Systems Appl. 10 (2001), no. 2, 201–213.

Jones, R.L.– New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem. Proc. Amer. Math. Soc.87 (1983), no. 4, 681–684.

Karlsson, A.– A proof of the subadditive ergodic theorem. Groups, graphs and random walks, 343–354, London Math. Soc. Lecture Note Ser., 436, CambridgeUniv. Press, Cambridge, 2017.

Karlsson, A.; Margulis, G.– A multiplicative ergodic theorem and nonpositively curved spaces. Comm. Math. Phys.208 (1999), no. 1, 107–123.

Katok, A.; Hasselblatt, B.–Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge Univ. Press, Cambridge, 1995.

Katznelson, Y.; Weiss, B.– A simple proof of some ergodic theorems.Israel J. Math.42 (1982), no. 4, 291–296.

Keane, M.; Petersen, K.– Easy and nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem. Dynamics & stochastics, 248–251. IMS Lecture Notes Monogr. Ser., 48. Inst. Math. Statist., Beachwood, OH, 2006.

Kingman, J.F.C.– The ergodic theory of subadditive stochastic processes.J. Roy. Statist.Soc. Ser. B, 30 (1968), 499–510.

Knill, O.– The upper Lyapunov exponent of SL(2,R) cocycles: discontinuity and the problem of positivity. Lyapunov exponents (Oberwolfach, 1990), 86–97, Lecture Notes inMath., 1486, Springer, Berlin, 1991.

Kornfeld, I.– Some old and new Rokhlin towers.Chapel Hill Ergodic Theory Workshops,145–169, Contemp. Math., 356, Amer. Math. Soc., Providence, RI, 2004.

Krengel, U.–Ergodic theorems. De Gruyter Studies in Mathematics, 6. Walter de Gruyter& Co., Berlin, 1985.

Lessa, P.–Teoremas ergódicos en espacios hiperbólicos. Master’s thesis, Universidad de la República, Montevideo (2009). ̃lessa/masters.html

Morris, I.D.– Mather sets for sequences of matrices and applications to the study of joint spectral radii. Proc. Lond. Math. Soc.107 (2013), no. 1, 121–150.

Petersen, K.–Ergodic theory. Corrected reprint of the 1983 original. Cambridge Studies in Advanced Mathematics, 2. Cambridge Univ. Press, Cambridge, 1989.

Quas, A.–

Ralston, D.– Heaviness: an extension of a lemma of Y. Peres. Houston J. Math.35 (2009),no. 4, 1131–1141.

Rokhlin, V.– A “general” measure-preserving transformation is not mixing. (Russian) Doklady Akad. Nauk SSSR60, (1948), 349–351.

Silva, C.E.; Thieullen, P.– The subadditive ergodic theorem and recurrence properties of Markovian transformations.J. Math. Anal. Appl. 154 (1991), no. 1, 83–99.

Steele, J.M.– Explaining a mysterious maximal inequality – and a path to the law of large numbers. Amer. Math. Monthly122 (2015), no. 5, 490–494.

Weiss, B.– On the work of V. A. Rokhlin in ergodic theory.Ergodic Theory Dynam. Systems9 (1989), no. 4, 619–627.

Wojtkowski, M.– Invariant families of cones and Lyapunov exponents.Ergodic Theory Dynam. Systems5 (1985), no. 1, 145–161.

How to Cite
Bochi, J. (2019). The basic ergodic theorems, yet again. CUBO, A Mathematical Journal, 20(3), 81–95.