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Hopf's Ergodic Theorem for Particles with Different Velocities and the "Strong Sweeping out Property"

  • A. Bellow (a1), A. P. Calderón (a2) and U. Krengel

Abstract

In an earlier paper we provided a counterexample to an old conjecture of Hopf. In this note we show that the "strong sweeping out property" obtains for the Hopf operators (Tt ) both when t —> +∞ and when t —> 0+, that is a.e. convergence fails in the worst possible way.

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Copyright

References

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1. Bellow, A. and Krengel, U., On Hopf's Ergodic Theorem for particles with different velocities, Almost Everywhere Convergence II, Academic Press, 1991, (eds. A. Bellow and R. Jones), 4147.
2. de Guzman, M., Real Variable Methods in Fourier Analysis, North-Holland Math. Stud. 46, 1981.
3. Hopf, E., Über die Bedeutung der willkürlichen Funktionen für die Wahrscheinlichkeitstheorie, Jahresber. Deutsch. Math.-Verein. 46(1936), 179195.
4. del, A. Junco and Rosenblatt, J., Counterexamples in Ergodic Theory and Number Theory, Math. Ann. 245(1979), 185197.
5. Krengel, U., Ergodic Theorems, de Gruyter Stud. Math. 6, 1985.
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Keywords

Hopf's Ergodic Theorem for Particles with Different Velocities and the "Strong Sweeping out Property"

  • A. Bellow (a1), A. P. Calderón (a2) and U. Krengel

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