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Ranks on the Baire class $ \xi$ functions


Authors: Márton Elekes, Viktor Kiss and Zoltán Vidnyánszky
Journal: Trans. Amer. Math. Soc. 368 (2016), 8111-8143
MSC (2010): Primary 26A21; Secondary 03E15, 54H05
DOI: https://doi.org/10.1090/tran/6764
Published electronically: April 14, 2016
MathSciNet review: 3546795
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Abstract: In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $ 1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $ \xi $ functions and generalize most of the results from the Baire class $ 1$ case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so-called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $ \omega _1$. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions.


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Additional Information

Márton Elekes
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary – and – Department of Analysis, Eötvös Loránd University, Pázmány P. s. 1/c, H-1117, Budapest, Hungary
Email: elekes.marton@renyi.mta.hu

Viktor Kiss
Affiliation: Department of Analysis, Eötvös Loránd University, Pázmány P. s. 1/c, H-1117, Budapest, Hungary
Email: kivi@cs.elte.hu

Zoltán Vidnyánszky
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary – and – Department of Analysis, Eötvös Loránd University, Pázmány P. s. 1/c, H-1117, Budapest, Hungary
Email: vidnyanszky.zoltan@renyi.mta.hu

DOI: https://doi.org/10.1090/tran/6764
Keywords: Baire class $\xi$ functions, ordinal ranks, descriptive set theory.
Received by editor(s): June 23, 2014
Received by editor(s) in revised form: May 23, 2015
Published electronically: April 14, 2016
Additional Notes: The first author was partially supported by the Hungarian Scientific Foundation grant no. 83726.
The second author was partially supported by the Hungarian Scientific Foundation grant no. 105645.
The third author was partially supported by the Hungarian Scientific Foundation grant no. 104178.
Article copyright: © Copyright 2016 American Mathematical Society