On density of infinite subsets I

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May 2019, 39(5): 2343-2359. doi: 10.3934/dcds.2019099

On density of infinite subsets I

Changguang Dong

Department of Mathematics, University of Maryland College Park, College Park, MD 20742, USA

Received June 2017 Revised July 2018 Published January 2019

Fund Project: Research was partially supported by NSF grant DMS 1602409.

Let

$ Y $

be a compact metric space,

$ G $

be a group acting by transformations on

$ Y $

. For any infinite subset

$ A\subset Y $

, we study the density of

$ gA $

for

$ g\in G $

and quantitative density of the set

$ {\bigcup\limits_{g\in G_n}gA} $

by the Hausdorff semimetric

$ d^H $

, for a family of increasing subsets

$ G_n\subset G $

. It is proven that for any integer

$ n\ge 2 $

,

$ \epsilon>0 $

, any infinite subset

$ A\subset \mathbb T^n $

, there is a

$ g\in SL(n,\mathbb Z) $

such that

$ gA $

is

$ \epsilon $

-dense. We also show that, for any infinite subset

$ A\subset [0,1] $

, for a.e. rotation and a.e. 3-IET,

$ \liminf\limits_nn\cdot d^H\left(\bigcup\limits_{k = 0}^{n-1}T^kA,[0,1]\right) = 0. $

Keywords: Density, Glasner property, group action, interval exchange.

Mathematics Subject Classification: Primary: 11B05, 37C20.

Citation: Changguang Dong. On density of infinite subsets I. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2343-2359. doi: 10.3934/dcds.2019099

References:

[1]	N. Alon and Y. Peres, Uniform dilations, Geom. Funct. Anal., 2 (1992), 1-28. doi: 10.1007/BF01895704.
[2]	Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅲ), Annals of Mathematics, 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5.
[3]	D. Berend and Y. Peres, Asymptotically dense dilations of sets on the circle, J. London Math. Soc., (2) 47 (1993), 1-17. doi: 10.1112/jlms/s2-47.1.1.
[4]	J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, Journal of the American Mathematical Society, 24 (2011), 231-280. doi: 10.1090/S0894-0347-2010-00674-1.
[5]	J. Bourgain, E. Lindenstrauss, Ph. Michel and A. Venkatesh, Some effective results for ×a; ×b, Ergodic Theory and Dynamical Systems, 29 (2009), 1705-1722. doi: 10.1017/S0143385708000898.
[6]	C. Dong, On density of infinite subsets Ⅱ: Dynamics on homogeneous spaces, to appear in Proceedings of the AMS. doi: 10.1090/proc/14298.
[7]	R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Math. J, 8 (1941), 243-255. doi: 10.1215/S0012-7094-41-00818-9.
[8]	S. Glasner, Almost periodic sets and measures on the torus, Israel J. Math., 32 (1979), 161-172. doi: 10.1007/BF02764912.
[9]	Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory and Dynamical Systems, 24 (2004), 767-802. doi: 10.1017/S0143385703000440.
[10]	Peter Humphries, http://mathoverflow.net/a/261553/99556
[11]	B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Annals of Mathematics, 174 (2011), 361-400. doi: 10.4007/annals.2011.174.1.10.
[12]	H. Kato, Continuum-wise expansive homeomorphisms, Can. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.
[13]	A. Katok and A. M. Stepin, Approximation of ergodic dynamical systems by periodic transformations, Soviet Math. Dokl, 7 (1966), 1638-1641.
[14]	A. Katok, S Katok and K. Schmidt, Rigidity of measurable structure for $\mathbb Z^d$ actions by automorphisms of a torus, Comm. Math. Helv., 77 (2002), 718-745. doi: 10.1007/PL00012439.
[15]	M. Kelly and T. Lê, Uniform dilations in higher dimensions, J. London Math. Soc., (2) 88 (2013), 925-940. doi: 10.1112/jlms/jdt054.
[16]	A. I. Khintchine, Continued Fractions, University of Chicago Press, 1964.
[17]	R. Nair and S. Velani, Glasner sets and polynomials in primes, Proceedings of the American Mathematical Society, 126 (1998), 2835-2840. doi: 10.1090/S0002-9939-98-04396-2.
[18]	J. Rodriguez Hertz, There are no stable points for continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598, https://arXiv.org/abs/math/0208102v3 doi: 10.4153/CJM-1993-030-4.
[19]	Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, International Mathematics Research Notices, (2010), 3744-3821. doi: 10.1093/imrn/rnq222.

show all references

References:

[1]	N. Alon and Y. Peres, Uniform dilations, Geom. Funct. Anal., 2 (1992), 1-28. doi: 10.1007/BF01895704.
[2]	Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (Ⅲ), Annals of Mathematics, 178 (2013), 1017-1059. doi: 10.4007/annals.2013.178.3.5.
[3]	D. Berend and Y. Peres, Asymptotically dense dilations of sets on the circle, J. London Math. Soc., (2) 47 (1993), 1-17. doi: 10.1112/jlms/s2-47.1.1.
[4]	J. Bourgain, A. Furman, E. Lindenstrauss and S. Mozes, Stationary measures and equidistribution for orbits of nonabelian semigroups on the torus, Journal of the American Mathematical Society, 24 (2011), 231-280. doi: 10.1090/S0894-0347-2010-00674-1.
[5]	J. Bourgain, E. Lindenstrauss, Ph. Michel and A. Venkatesh, Some effective results for ×a; ×b, Ergodic Theory and Dynamical Systems, 29 (2009), 1705-1722. doi: 10.1017/S0143385708000898.
[6]	C. Dong, On density of infinite subsets Ⅱ: Dynamics on homogeneous spaces, to appear in Proceedings of the AMS. doi: 10.1090/proc/14298.
[7]	R. J. Duffin and A. C. Schaeffer, Khintchine's problem in metric Diophantine approximation, Duke Math. J, 8 (1941), 243-255. doi: 10.1215/S0012-7094-41-00818-9.
[8]	S. Glasner, Almost periodic sets and measures on the torus, Israel J. Math., 32 (1979), 161-172. doi: 10.1007/BF02764912.
[9]	Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory and Dynamical Systems, 24 (2004), 767-802. doi: 10.1017/S0143385703000440.
[10]	Peter Humphries, http://mathoverflow.net/a/261553/99556
[11]	B. Kalinin, A. Katok and F. Rodriguez Hertz, Nonuniform measure rigidity, Annals of Mathematics, 174 (2011), 361-400. doi: 10.4007/annals.2011.174.1.10.
[12]	H. Kato, Continuum-wise expansive homeomorphisms, Can. J. Math., 45 (1993), 576-598. doi: 10.4153/CJM-1993-030-4.
[13]	A. Katok and A. M. Stepin, Approximation of ergodic dynamical systems by periodic transformations, Soviet Math. Dokl, 7 (1966), 1638-1641.
[14]	A. Katok, S Katok and K. Schmidt, Rigidity of measurable structure for $\mathbb Z^d$ actions by automorphisms of a torus, Comm. Math. Helv., 77 (2002), 718-745. doi: 10.1007/PL00012439.
[15]	M. Kelly and T. Lê, Uniform dilations in higher dimensions, J. London Math. Soc., (2) 88 (2013), 925-940. doi: 10.1112/jlms/jdt054.
[16]	A. I. Khintchine, Continued Fractions, University of Chicago Press, 1964.
[17]	R. Nair and S. Velani, Glasner sets and polynomials in primes, Proceedings of the American Mathematical Society, 126 (1998), 2835-2840. doi: 10.1090/S0002-9939-98-04396-2.
[18]	J. Rodriguez Hertz, There are no stable points for continuum-wise expansive homeomorphisms, Canad. J. Math., 45 (1993), 576-598, https://arXiv.org/abs/math/0208102v3 doi: 10.4153/CJM-1993-030-4.
[19]	Z. Wang, Quantitative density under higher rank abelian algebraic toral actions, International Mathematics Research Notices, (2010), 3744-3821. doi: 10.1093/imrn/rnq222.

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