Divergence of averages obtained by sampling a flow

Authors:
Mustafa Akcoglu, Alexandra Bellow, Andrés del Junco and Roger L. Jones

Journal:
Proc. Amer. Math. Soc. **118** (1993), 499-505

MSC:
Primary 28D10; Secondary 47A35, 58F11

MathSciNet review:
1143221

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider ergodic averages obtained by sampling at discrete times along a measure preserving ergodic flow. We show, in particular, that if is an aperiodic flow, then averages obtained by sampling at times satisfy the strong sweeping out property for any sequence . We also show that there is a flow (which is periodic) and a sequence such that the Cesaro averages of samples at time do converge a.e. In fact, we show that every uniformly distributed sequence admits a perturbation that makes it a good Lebesgue sequence.

**[1]**M. A. Akcoglu, A. del Junco, and W. M. F. Lee,*A solution to a problem of A. Bellow*, Almost everywhere convergence, II (Evanston, IL, 1989) Academic Press, Boston, MA, 1991, pp. 1–7. MR**1131778****[2]**Dietrich Kölzow (ed.),*Measure theory*, Lecture Notes in Mathematics, Vol. 541, Springer-Verlag, Berlin-New York, 1976. MR**0435324****[3]**M. Akcoglu, A. Bellow, R. Jones, V. Losert, K. Reinhold, and M. Wierdl,*The strong sweeping out property for lacunary sequences, for the Riemann sums, and related matters*, preprint.**[4]**V. Bergelson, M. Boshernitzan, and J. Bourgain,*Some results on non-linear recurrence*, preprint.**[5]**J. Bourgain,*Almost sure convergence and bounded entropy*, Israel J. Math.**63**(1988), no. 1, 79–97. MR**959049**, 10.1007/BF02765022**[6]**Andrés del Junco and Joseph Rosenblatt,*Counterexamples in ergodic theory and number theory*, Math. Ann.**245**(1979), no. 3, 185–197. MR**553340**, 10.1007/BF01673506**[7]**Roger L. Jones and Máté Wierdl,*Convergence and divergence of ergodic averages*, Ergodic Theory Dynam. Systems**14**(1994), no. 3, 515–535. MR**1293406**, 10.1017/S0143385700008002**[8]**D. A. Lind,*Locally compact measure preserving flows*, Advances in Math.**15**(1975), 175–193. MR**0382595****[9]**Joseph Rosenblatt,*Universally bad sequences in ergodic theory*, Almost everywhere convergence, II (Evanston, IL, 1989) Academic Press, Boston, MA, 1991, pp. 227–245. MR**1131794**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
28D10,
47A35,
58F11

Retrieve articles in all journals with MSC: 28D10, 47A35, 58F11

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1993-1143221-1

Keywords:
Ergodic flow,
good Lebesgue sequences,
uniform distribution

Article copyright:
© Copyright 1993
American Mathematical Society