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Transactions of the American Mathematical Society

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On measure-preserving $ \mathcal{C}^1$ transformations of compact-open subsets of non-archimedean local fields


Authors: James Kingsbery, Alex Levin, Anatoly Preygel and Cesar E. Silva
Journal: Trans. Amer. Math. Soc. 361 (2009), 61-85
MSC (2000): Primary 37A05; Secondary 37F10
DOI: https://doi.org/10.1090/S0002-9947-08-04686-2
Published electronically: August 12, 2008
MathSciNet review: 2439398
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Abstract: We introduce the notion of a locally scaling transformation defined on a compact-open subset of a non-archimedean local field. We show that this class encompasses the Haar measure-preserving transformations defined by $ \mathcal{C}^1$ (in particular, polynomial) maps, and prove a structure theorem for locally scaling transformations. We use the theory of polynomial approximation on compact-open subsets of non-archimedean local fields to demonstrate the existence of ergodic Markov, and mixing Markov transformations defined by such polynomial maps. We also give simple sufficient conditions on the Mahler expansion of a continuous map $ \mathbb{Z}_p \to \mathbb{Z}_p$ for it to define a Bernoulli transformation.


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Additional Information

James Kingsbery
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: 06jck@williams.edu

Alex Levin
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, MIT, Cambridge, Massachusetts 02139
Email: alex.levin@post.harvard.edu, levin@mit.edu

Anatoly Preygel
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, MIT, Cambridge, Massachusetts 02139
Email: preygel@post.harvard.edu, preygel@mit.edu

Cesar E. Silva
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: csilva@williams.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04686-2
Keywords: Measure-preserving, ergodic, non-archimedean local field
Received by editor(s): September 1, 2006
Published electronically: August 12, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.