Abstract
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.
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Eigen, S., Hajian, A., Ito, Y., Prasad, V. (2014). Isomorphism Invariants. In: Weakly Wandering Sequences in Ergodic Theory. Springer Monographs in Mathematics. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55108-9_6
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DOI: https://doi.org/10.1007/978-4-431-55108-9_6
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