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recurrence for dynamical systems; non-recurrence for dynamical systems; rotations of the unit circle; syndetic set; Bohr topology on $\mathbb {Z}$; Bohr set; $r$-Bohr set
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb T$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb T$, and Bohr if it is recurrent for all products of rotations on $\mathbb T$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb Z$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.
[1] Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211 (2007), 766-793. DOI 10.1016/j.aim.2006.09.010 | MR 2323544 | Zbl 1123.47006
[2] Bergelson, V., Junco, A. Del, Lemańczyk, M., Rosenblatt, J.: Rigidity and non-recurrence along sequences. Preprint 2011, arXiv:1103.0905.
[3] Bergelson, V., Haland, I. J.: Sets of recurrence and generalized polynomials. V. Bergelson, et al. Convergence in Ergodic Theory and Probability Papers from the conference, Ohio State University, Columbus, OH, USA, June 23-26, 1993, de Gruyter, Berlin, Ohio State Univ. Math. Res. Inst. Publ. 5 (1996), 91-110. MR 1412598 | Zbl 0958.28014
[4] Bergelson, V., Lesigne, E.: Van der Corput sets in $\mathbb Z^d$. Colloq. Math. 110 (2008), 1-49. DOI 10.4064/cm110-1-1 | MR 2353898
[5] Boshernitzan, M., Glasner, E.: On two recurrence problems. Fundam. Math. 206 (2009), 113-130. DOI 10.4064/fm206-0-7 | MR 2576263 | Zbl 1187.37020
[6] Furstenberg, H.: Poincaré recurrence and number theory. Bull. Am. Math. Soc., New Ser. 5 (1981), 211-234. DOI 10.1090/S0273-0979-1981-14932-6 | MR 0628658 | Zbl 0481.28013
[7] Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton, N. J. (1981). MR 0603625 | Zbl 0459.28023
[8] Glasner, E.: On minimal actions of Polish groups. Topology Appl. 85 (1998), 119-125. DOI 10.1016/S0166-8641(97)00143-0 | MR 1617456 | Zbl 0923.54030
[9] Glasner, E.: Classifying dynamical systems by their recurrence properties. Topol. Methods Nonlinear Anal. 24 (2004), 21-40. MR 2111980 | Zbl 1072.37017
[10] Glasner, E., Weiss, B.: On the interplay between measurable and topological dynamics. Handbook of Dynamical Systems vol. 1B B. Hasselblatt et al. Amsterdam, Elsevier (2006), 597-648. MR 2186250 | Zbl 1130.37303
[11] Grivaux, S.: Non-recurrence sets for weakly mixing dynamical systems. (to appear) in Ergodic Theory Dyn. Syst., DOI 10.1017/etds.2012.116
[12] Kannan, R., Lovász, L.: Covering minima and lattice-point-free convex bodies. Ann. Math. (2) 128 (1988), 577-602. MR 0970611 | Zbl 0659.52004
[13] Katznelson, Y.: Chromatic numbers of Cayley graphs on $\mathbb Z$ and recurrence. Combinatorica 21 (2001), 211-219. DOI 10.1007/s004930100019 | MR 1832446
[14] Kříž, I.: Large independent sets in shift-invariant graphs. Graphs Comb. 3 (1987), 145-158. DOI 10.1007/BF01788538 | MR 0932131 | Zbl 0641.05044
[15] Pestov, V.: Forty-plus annotated questions about large topological groups. Open Problems in Topology II Elliott M. Pearl Elsevier, Amsterdam (2007), 439-450.
[16] Petersen, K.: Ergodic Theory. Cambridge Studies in Advanced Mathematics 2 Cambridge University Press, Cambridge (1983). MR 0833286 | Zbl 0507.28010
[17] Queffélec, M.: Substitution Dynamical Systems. Spectral analysis. 2nd ed. Lecture Notes in Mathematics 1294. Springer, Berlin (2010). MR 2590264 | Zbl 1225.11001
[18] Veech, W. A.: The equicontinuous structure relation for minimal abelian transformation groups. Am. J. Math. 90 (1968), 723-732. DOI 10.2307/2373480 | MR 0232377 | Zbl 0177.51204
[19] Walters, P.: An Introduction to Ergodic Theory. Graduate Texts in Mathematics vol. 79 Springer, New-York (1982). DOI 10.1007/978-1-4612-5775-2 | MR 0648108 | Zbl 0475.28009
[20] Weiss, B.: Single Orbit Dynamics. CBMS Regional Conference Series in Math. 95. AMS, Providence (2000). MR 1727510 | Zbl 1083.37500
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