Isomorphism Invariants

  • Stanley Eigen
  • Arshag Hajian
  • Yuji Ito
  • Vidhu Prasad
Part of the Springer Monographs in Mathematics book series (SMM)


In this chapter we examine several isomorphism invariants associated to an infinite ergodic transformation. We use these invariants to show the non-isomorphism of various infinite ergodic transformations. In Sect. 6.1 the role of eww sets and sequences is discussed and used to distinguish between some infinite ergodic transformations. In Sect. 6.2 it is shown that there exists a class of infinite ergodic transformations that exhibit a regularity in the size of the return sets of finite measure. We define (an isomorphism invariant) the α-type for such transformations. Previously, in Sect. 3.3 recurrent sequences for an infinite ergodic transformation were defined, and then in Sect. 4.1 the First Basic Example was constructed and all its recurrent sequences were computed. This is extended in Sect. 6.3 to a family of transformations where all the recurrent sequences for the members of that family are computed and shown to distinguish between any two members of the family. Finally, in Sect. 6.4 the growth distribution of ww sets is also shown to be an effective isomorphism invariant.


Finite Type Finite Measure Space Interval Recurrent Sequence Growth Distribution 
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  1. 1.
    Aaronson, J.: An Introduction to Infinite Ergodic Theory. AMS Mathematical Surveys and Monographs, vol. 50, xii+284 pp. American Mathematical Society, Providence (1997)Google Scholar
  2. 18.
    Eigen, S., Hajian, A., Prasad, V.: Universal skyscraper templates for infinite measure preserving transformations. Discrete Continuous Dyn. Syst. 16, 343–360 (2006)MATHMathSciNetGoogle Scholar
  3. 21.
    Friedman, N.: Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, vol. 29, v+143 pp. Van Nostrand Reinhold, New York (1970)Google Scholar
  4. 30.
    Hajian, A., Ito, Y.: Transformations that do not accept a finite invariant measue. Bull. Am. Math. Soc. 84, 417–427 (1978)MATHMathSciNetGoogle Scholar
  5. 35.
    Hamachi, T., Osikawa, M.: On zero type and positive type transformations with infinite invariant measures. Mem. Fac. Sci. Kyushu Univ. Ser. A 25, 280–295 (1971)MATHMathSciNetGoogle Scholar
  6. 44.
    Kakutani, S.: Classification of ergodic groups of automorphisms. In: Proceedings of the International Conference on Functional Analysis and Related Topics, pp. 392–397, Tokyo, April 1969Google Scholar
  7. 51.
    Ornstein, D.S., Shields, P.C.: An uncountable family of K-automorphisms. Adv. Math. 10, 63–88 (1973)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  • Stanley Eigen
    • 1
  • Arshag Hajian
    • 1
  • Yuji Ito
    • 2
  • Vidhu Prasad
    • 3
  1. 1.Department of MathematicsNortheastern UniversityBostonUSA
  2. 2.Department of MathematicsKeio UniversityYokohamaJapan
  3. 3.Department of Mathematical SciencesUniversity of Massachusetts LowellLowellUSA

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