Abstract
The main purpose of this paper is to prove the following local ergodic theorem. If {T(t1,…,tN): ti ≥ 0, i=1,…,N} is an N-parameter strongly continuous semigroup of positive contraction operators on L1(Θ,F,μ), then for all f є L1(Θ) \(M\left( \varepsilon \right)f = \frac{1}{{_\varepsilon N}}\int_o^\varepsilon \ldots \int_o^\varepsilon {T\left( {t_1 , \ldots ,t_N } \right)f dt_1 } \ldots dt_N\) converges almost everywhere as ε + 0 + . The theorem will also be proved for semigroups of non-positive L1 contractions which are also L∞ contracting. Examples are given to show that the maximal ergodic lemmas which are used to prove the one parameter case of the above results do not extend to the N-parameter case. The N-parameter case is proved by successive reduction of the number of parameters.
Keywords
- Ergodic Theorem
- Continuous Semigroup
- Contraction Operator
- Norm Topology
- Strong Operator Topology
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The results of this paper are to appear in the author's Ph.D. thesis, which is being written under the direction of Professor U. Krengel.
Part of this work has been supported through N.S.F. grant GP 9354.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Akcoglu, M. A. and R. V. Chacon: A Local Ration Theorem. to appear.
Busemann, H. and W. Feller: Zur Differentiation der Lebesgueshen Integrale. Fund. Math. 22, 226–256 (1934).
Dunford, N. and J. T. Schwartz: Convergence almost everywhere of operator averages. J. Rational Mech. and Anal. 5, 129–178 (1956).
Krengel, U.: A Local Ergodic Theorem. Inventiones Math. 6, 329–333 (1969).
Ornstein, D.: The Sums of Iterates of a Positive Operator. to appear.
Wiener, N.: The Ergodic Theorem. Duke Math. J. 5, 1–18 (1939).
Rights and permissions
Copyright information
© 1970 Springer-Verleg
About this paper
Cite this paper
Terrell, T.R. (1970). Local ergodic theorems for N-parameter semigroups of operators. In: Contributions to Ergodic Theory and Probability. Lecture Notes in Mathematics, vol 160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0060658
Download citation
DOI: https://doi.org/10.1007/BFb0060658
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-05188-6
Online ISBN: 978-3-540-36371-2
eBook Packages: Springer Book Archive
