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Classifying Borel automorphisms

Published online by Cambridge University Press:  12 March 2014

John D. Clemens
John D. Clemens*
Affiliation:
Penn State University, Dept. of Mathematics, University Park, PA 16802, USA. E-mail: clemens@math.psu.edu
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Extract

§1. Introduction. This paper considers several complexity questions regarding Borel automorphisms of a Polish space. Recall that a Borel automorphism is a bijection of the space with itself whose graph is a Borel set (equivalently, the inverse image of any Borel set is Borel). Since the inverse of a Borel automorphism is another Borel automorphism, as is the composition of two Borel automorphisms, the set of Borel automorphisms of a given Polish space forms a group under the operation of composition. We can also consider the class of automorphisms of all Polish spaces. We will be primarily concerned here with the following notion of equivalence:

Definition 1.1. Two Borel automorphisms f and g of the Polish spaces X and Y are said to be Borel isomorphic, f ≅ g, if they are conjugate, i.e. there is a Borel bijection φ: X → Y such that φ ∘ f = g ∘ φ.

We restrict ourselves to automorphisms of uncountable Polish spaces, as the Borel automorphisms of a countable space are simply the permutations of the space. Since any two uncountable Polish spaces are Borel isomorphic, any Borel automorphism is Borel isomorphic to some automorphism of a fixed space. Hence, up to Borel isomorphism we can fix a Polish space and represent any Borel automorphism as an automorphism of this space. We will use the Cantor space 2ω (with the product topology) as our representative space.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 4 , December 2007 , pp. 1081 - 1092
Copyright
Copyright © Association for Symbolic Logic 2007

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References

REFERENCES

[1]Adams, S. and Kechris, A. S., Linear algebraic groups and countable Borel equivalence relations, Journal of the American Mathematical Society, vol. 13 (2000), no. 4, pp. 909–943.CrossRefGoogle Scholar
[2]Dougherty, R., Jackson, S., and Kechris, A. S., The structure of hyperfinite Borel equivalence relations, Transactions of the American Mathematical Society, vol. 341 (1994), no. 1, pp. 193–225.CrossRefGoogle Scholar
[3]Eigen, S., Hajian, A., and Weiss, B., Borel automorphisms with no finite invariant measure, Proceedings of the American Mathematical Society, vol. 126 (1998), no. 12, pp. 3619–3623.CrossRefGoogle Scholar
[4]Gao, S., Some applications of the Adams-Kechris technique, Proceedings of the American Mathematical Society, vol. 130 (2002), no. 3, pp. 863–874.CrossRefGoogle Scholar
[5]Hjorth, G., On invariants for measure preserving transformations, Fundamenta Mathematicae, vol. 169 (2001), no. 1, pp. 51–84.CrossRefGoogle Scholar
[6]Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, Springer-Verlag, Berlin, 1995.CrossRefGoogle Scholar
[7]Kechris, A. S. and Louveau, A., The classification of hypersmooth Borel equivalence relations, Journal of the American Mathematical Society, vol. 10 (1997), no. 1, pp. 215–242.CrossRefGoogle Scholar
[8]Miller, B. and Rosendal, C., Descriptive Kakutani equivalence, preprint.Google Scholar