Rudolph’s two step coding theorem and Alpern’s lemma for ℝ^{𝕕} actions

Kra, Bryna; Quas, Anthony; Şahi̇n, Ayşe

doi:10.1090/S0002-9947-2014-06247-8

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

Trans. Amer. Math. Soc. 367 (2015), 4253-4285 Request permission

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

Steve Alpern, Return times and conjugates of an antiperiodic transformation, Ergodic Theory Dynam. Systems 1 (1981), no. 2, 135–143. MR 661814, DOI 10.1017/s0143385700009172
Warren Ambrose and Shizuo Kakutani, Structure and continuity of measurable flows, Duke Math. J. 9 (1942), 25–42. MR 5800
J. P. Conze, Entropie d’un groupe abélien de transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 11–30 (French). MR 335754, DOI 10.1007/BF00533332
R. M. Dudley, Distances of probability measures and random variables, Ann. Math. Statist. 39 (1968), 1563–1572. MR 230338, DOI 10.1007/978-1-4419-5821-1_{4}
Mihail Kolountzakis, Lattice tilings by cubes: whole, notched and extended, Electron. J. Combin. 5 (1998), Research Paper 14, 11. MR 1605065
D. A. Lind, Locally compact measure preserving flows, Advances in Math. 15 (1975), 175–193. MR 382595, DOI 10.1016/0001-8708(75)90133-4
Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI 10.1090/S0273-0979-1980-14702-3
A. A. Prikhod′ko, Partitions of the phase space of a measure-preserving $\mathbf Z^d$-action into towers, Mat. Zametki 65 (1999), no. 5, 712–725 (Russian, with Russian summary); English transl., Math. Notes 65 (1999), no. 5-6, 598–609. MR 1716239, DOI 10.1007/BF02743170
E. Arthur Robinson Jr. and Ayşe A. Şahin, On the existence of Markov partitions for $\Bbb Z^d$ actions, J. London Math. Soc. (2) 69 (2004), no. 3, 693–706. MR 2050041, DOI 10.1112/S0024610703005064
Daniel Rudolph, A two-valued step coding for ergodic flows, Math. Z. 150 (1976), no. 3, 201–220. MR 414825, DOI 10.1007/BF01221147
Daniel J. Rudolph, Rectangular tilings of $\textbf {R}^n$ and free $\textbf {R}^n$-actions, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 653–688. MR 970578, DOI 10.1007/BFb0082854
Daniel J. Rudolph, Markov tilings of $\textbf {R}^n$ and representations of $\textbf {R}^n$ actions, Measure and measurable dynamics (Rochester, NY, 1987) Contemp. Math., vol. 94, Amer. Math. Soc., Providence, RI, 1989, pp. 271–290. MR 1012996, DOI 10.1090/conm/094/1012996
V. V. Ryzhikov, The Rohlin-Halmos property without $\epsilon$ does not hold for the actions of the group $\textbf {Z}^2$, Mat. Zametki 44 (1988), no. 2, 208–215, 287 (Russian); English transl., Math. Notes 44 (1988), no. 1-2, 596–600 (1989). MR 969270, DOI 10.1007/BF01159255
V. V. Ryzhikov, An estimate for the entropy of a class of $\textbf {Z}^2$-actions, Mat. Zametki 51 (1992), no. 2, 152–155 (Russian); English transl., Math. Notes 51 (1992), no. 1-2, 215–218. MR 1165481, DOI 10.1007/BF02102134
Ayşe A. Şahın, The $\Bbb Z^d$ Alpern multi-tower theorem for rectangles: a tiling approach, Dyn. Syst. 24 (2009), no. 4, 485–499. MR 2573000, DOI 10.1080/14689360903127188
Sherman Stein, The notched cube tiles $\textbf {R}^n$, Discrete Math. 80 (1990), no. 3, 335–337. MR 1049254, DOI 10.1016/0012-365X(90)90252-D
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423–439. MR 177430, DOI 10.1214/aoms/1177700153
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
A. Yu. Zaĭtsev, Estimates for the Lévy-Prokhorov distance in the multidimensional central limit theorem for random vectors with finite exponential moments, Teor. Veroyatnost. i Primenen. 31 (1986), no. 2, 246–265 (Russian). MR 850986