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Multiple recurrence and infinite measure preserving odometers

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Abstract

A family of infinite measure preserving odometers is presented which exhibit examples ofp-recurrent but notp+1-recurrent ergodic transformations for everyp>1.

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References

  1. J. Aaronson and M. Nadkarni,L eigenvalues and L 2 spectra of non-singular transformations, Proceedings of the London Mathematical Society55 (1987), 538–570.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Choksi, J. Hawkins and V. Prasad,Abelian cocyles for nonsingular ergodic transformations and genericity of type III 1 transformations, Monatschefte für Mathematik103 (1987), 187–205.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. G. De Bruijn,On bases for sets of integers, Publicationes Mathematicae Debrecen1 (1950), 232–242.

    MATH  MathSciNet  Google Scholar 

  4. S. Eigen and A. Hajian,Poincaré Sequences in Infinite Measure Spaces and Complementing Subsets of the Integers, Lecture Notes in Mathematics1342, Springer-Verlag, Berlin, 1988, pp. 154–157.

    Google Scholar 

  5. S. Eigen and A. Hajian,Sequences of integers and ergodic transformations, Annals of Mathematics73 (1989), 256–262.

    MATH  MathSciNet  Google Scholar 

  6. N. Friedman,Introduction to Ergodic Theory, Van Nostrand Reinhold, New York, 1970.

    MATH  Google Scholar 

  7. H. Furstenberg,Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton University Press, Princeton, N.J., 1981.

    MATH  Google Scholar 

  8. A. Hajian and S. Kakutani,An example of an ergodic m.p.t. defined on an infinite measure space, Lecture Notes in Mathematics160, Springer-Verlag, Berlin, 1970, pp. 45–52.

    Google Scholar 

  9. T. Hamachi and M. Osikawa,Ergodic groups of automorphisms and Krieger’s theoerems, Seminar on Mathematical Sciences, Keio University, 1981.

  10. D. Hill,σ-finite invariant measures on infinite product spaces, Transactions of the American Mathematical Society153 (1971), 347–370.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Host, J. F. Mela and F. Parreau,Nonsingular transformations and spectral analysis of measures, Bulletin de la Société Mathématique de France119 (1991), 33–90.

    MATH  MathSciNet  Google Scholar 

  12. Y. Ito, T. Kamae and J. Shiokawa,Point spectrum and Hausdorff dimension, inNuclear Theory and Combinatorics (J. Akiyama et al., eds.), World Scientific, Singapore, 1985.

    Google Scholar 

  13. D. Rudolph and C. Silva,Minimal self-joinings for nonsingular transformations, Ergodic Theory and Dynamical Systems9 (1989), 759–800.

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Schmidt,Some solved and unsolved problems concerning orbit equivalence of countable group actions, inProceedings of the Conference on Ergodic Theory and Related Topics II (Georgenthal, 1986), Teubner-Texte Math 94, Teubner, Leipzig, 1987, pp. 171–184.

    Google Scholar 

  15. E. Szemerédi,On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica27 (1975), 199–245.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Stanley Eigen.

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Eigen, S., Hajian, A. & Halverson, K. Multiple recurrence and infinite measure preserving odometers. Israel J. Math. 108, 37–44 (1998). https://doi.org/10.1007/BF02783041

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  • DOI: https://doi.org/10.1007/BF02783041

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