Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T04:59:47.670Z Has data issue: false hasContentIssue false
  • English
  • Français

A ratio ergodic theorem for Borel actions of ℤd×ℝk

Published online by Cambridge University Press:  16 January 2012

ERIC HOLT
ERIC HOLT*
Affiliation:
Mathematics & Science Division, North Lake College, Irving, TX 75038, USA (email: eholt@dcccd.edu)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove a ratio ergodic theorem for free Borel actions of ℤd×ℝk on a standard Borel σ-finite measure space. The proof employs a lemma by Hochman involving coarse dimension, as well as the Besicovitch covering lemma. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore use Rudolph’s diffusion of the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form (1/(μx(Bn)))∫ Bnf∘T−v(x) dμx(v). A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 2: Daniel J. Rudolph – in Memoriam , April 2012 , pp. 675 - 689
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Besicovitch, A.. A general form of the covering principle and relative differentiation of additive functions. Proc. Cambridge Philos. Soc. 41 (1945), 103–110.CrossRefGoogle Scholar
[2]Birkhoff, G.. Proof of the ergodic theorem. Proc. Natl. Acad. Sci. USA 17 (1931), 656–660.CrossRefGoogle ScholarPubMed
[3]Feldman, J.. A ratio ergodic theorem for commuting, conservative, invertible transformations with quasi-invariant measure summed over symmetric hypercubes. Ergod. Th. & Dynam. Sys. 27(4) (2007), 1135–1142.CrossRefGoogle Scholar
[4]Hochman, M.. A ratio ergodic theorem for multiparameter non-singular actions. J. Eur. Math. Soc. 12(2) (2010), 365–383.CrossRefGoogle Scholar
[5]Hopf, E.. Ergodentheorie. J. Springer, Berlin, 1937.CrossRefGoogle Scholar
[6]Hurewicz, W.. Ergodic theorem without invariant measure. Ann. of Math. (2) 45(1) (1944), 192–206.CrossRefGoogle Scholar
[7]Krengel, U., (with supplement by Antoine Brunel). Ergodic Theorems (de Gruyter Studies in Mathematics, 6). Walter de Gruyter & Co., Berlin, 1985.CrossRefGoogle Scholar
[8]Einsiedler, M. and Lindenstrauss, E.. Diagonal actions on locally homogeneous spaces. Homogeneous Flows, Moduli Spaces and Arithmetic (Clay Mathematics Proceedings, 10). American Mathematical Society, Providence, RI, 2010, pp. 155–241.Google Scholar
[9]Oxtoby, J.. On the ergodic theorem of Hurewicz. Ann. of Math. (2) 49(4) (1948), 872–884.CrossRefGoogle Scholar
[10]Rudolph, D.. Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold. Ergod. Th. & Dynam. Sys. 2 (1982), 491–512.CrossRefGoogle Scholar
[11]Rudolph, D.. Ergodic theory on Borel foliations by ℝn and ℤn. Contemp. Math. 444 (2007), 89–113.CrossRefGoogle Scholar
[12]Shields, P.. The ergodic and entropy theorems revisited. IEEE Trans. Inform. Theory 33(2) (1987), 263–266.CrossRefGoogle Scholar