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Rudolph's two step coding theorem and Alpern's lemma for $ \mathbb{R}^d$ actions

Authors: Bryna Kra, Anthony Quas and Ayşe Şahin
Journal: Trans. Amer. Math. Soc. 367 (2015), 4253-4285
MSC (2010): Primary 37A15; Secondary 37B50
Published electronically: October 1, 2014
MathSciNet review: 3324927
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Abstract: Rudolph showed that the orbits of any measurable, measure preserving $ \mathbb{R}^d$ action can be measurably tiled by $ 2^d$ rectangles and asked if this number of tiles is optimal for $ d>1$. In this paper, using a tiling of $ \mathbb{R}^d$ by notched cubes, we show that $ d+1$ tiles suffice. Furthermore, using a detailed analysis of the set of invariant measures on tilings of $ \mathbb{R}^2$ by two rectangles, we show that while for $ \mathbb{R}^2$ actions with completely positive entropy this bound is optimal, there exist mixing $ \mathbb{R}^2$ actions whose orbits can be tiled by 2 tiles.

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Additional Information

Bryna Kra
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208

Anthony Quas
Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada

Ayşe Şahin
Affiliation: Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Avenue, Chicago, Illinois 60614

Received by editor(s): November 20, 2012
Received by editor(s) in revised form: July 8, 2013
Published electronically: October 1, 2014
Additional Notes: The first author was partially supported by NSF grant $1200971$
The second author was partially supported by NSERC
Dedicated: Dedicated to the memory of Daniel J. Rudolph
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.