Rudolph’s two step coding theorem and Alpern’s lemma for $\mathbb {R}^d$ actions
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- by Bryna Kra, Anthony Quas and Ayşe Şahi̇n PDF
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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.References
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Additional Information
- Bryna Kra
- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 363208
- ORCID: 0000-0002-5301-3839
- Email: kra@math.northwestern.edu
- Anthony Quas
- Affiliation: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada
- MR Author ID: 317685
- Email: aquas@uvic.ca
- Ayşe Şahi̇n
- Affiliation: Department of Mathematical Sciences, DePaul University, 2320 N. Kenmore Avenue, Chicago, Illinois 60614
- Email: asahin@depaul.edu
- 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 - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 4253-4285
- MSC (2010): Primary 37A15; Secondary 37B50
- DOI: https://doi.org/10.1090/S0002-9947-2014-06247-8
- MathSciNet review: 3324927
Dedicated: Dedicated to the memory of Daniel J. Rudolph