Topological mild mixing of all orders along polynomials

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March 2022, 42(3): 1163-1184. doi: 10.3934/dcds.2021150

Topological mild mixing of all orders along polynomials

Yang Cao and Song Shao ^,

CAS Wu Wen-Tsun Key Laboratory of Mathematics, Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China

^*Corresponding author: Song Shao

Received March 2021 Revised August 2021 Published March 2022 Early access November 2021

Fund Project: This research is supported by NNSF of China (11971455, 11571335)

A minimal system $ (X,T) $ is topologically mildly mixing if for all non-empty open subsets $ U,V $, $ \{n\in {\mathbb Z}: U\cap T^{-n}V\neq \emptyset\} $ is an IP$ ^* $-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that $ (X,T) $ is a topologically mildly mixing minimal system, $ d\in {\mathbb N} $, $ p_1(n),\ldots, p_d(n) $ are integral polynomials with no $ p_i $ and no $ p_i-p_j $ constant, $ 1\le i\neq j\le d $. Then for all non-empty open subsets $ U , V_1, \ldots, V_d $, $ \{n\in {\mathbb Z}: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \} $ is an IP$ ^* $-set. We also give the corresponding theorem for systems under abelian group actions.

Keywords: IP system, mild mixing, polynomial.

Mathematics Subject Classification: Primary: 37B20; Secondary: 37B05, 37A25.

Citation: Yang Cao, Song Shao. Topological mild mixing of all orders along polynomials. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1163-1184. doi: 10.3934/dcds.2021150

References:

[1]	D. Berend and V. Bergelson, Jointly ergodic measure-preserving transformations, Israel J. Math., 49 (1984), 307-314. doi: 10.1007/BF02760955.
[2]	V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349. doi: 10.1017/S0143385700004090.
[3]	V. Bergelson, Combinatorial and Diophantine applications of ergodic theory, Handbook of Dynamical Systems, 1 (2006), 745-869. doi: 10.1016/S1874-575X(06)80037-8.
[4]	V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753. doi: 10.1090/S0894-0347-96-00194-4.
[5]	V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales–Jewett theorem, Ann. of Math., 150 (1999), 33-75. doi: 10.2307/121097.
[6]	V. Bergelson, A. Leibman and Y. Son, Joint ergodicity along generalized linear functions, Ergodic Theory Dynam. Systems, 36 (2016), 2044-2075. doi: 10.1017/etds.2015.11.
[7]	V. Bergelson and R. McCutcheon, An ergodic IP polynomial Szemerédi theorem, Mem. Amer. Math. Soc., 146 (2000), 106pp. doi: 10.1090/memo/0695.
[8]	S. Donoso, A. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, J. Anal. Math., arXiv: 1902.10237
[9]	N. Frantzikinakis and B. Kra, Polynomial averages converge to the product of integrals, Israel J. Math., 148 (2005), 267-276. doi: 10.1007/BF02775439.
[10]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N. J., 1981.
[11]	H. Furstenberg, IP-systems in Ergodic Theory, Contemp. Math., 26 (1984), 131-148. doi: 10.1090/conm/026/737395.
[12]	H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Anal. Math., 45 (1985), 117-168. doi: 10.1007/BF02792547.
[13]	H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 668 (1978), 127-132.
[14]	H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Anal. Math., 34 (1978), 61-85. doi: 10.1007/BF02790008.
[15]	E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241-262. doi: 10.1007/BF03008411.
[16]	E. Glasner, W. Huang, S. Shao, B. Weiss and X. Ye, Topological characteristic factors and nilsystems, arXiv: 2006.12385.
[17]	E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical Systems, 1 (2006), 597-648. doi: 10.1016/S1874-575X(06)80035-4.
[18]	N. Hindman, Finite sums from sequences within cells of a partition of ${\mathbb N}$, J. Combinatorial Theory Ser. A, 17 (1974), 1-11. doi: 10.1016/0097-3165(74)90023-5.
[19]	W. Huang, S. Shao and X. Ye, Topological correspondence of multiple ergodic averages of nilpotent group actions, J. Anal. Math., 138 (2019), 687-715. doi: 10.1007/s11854-019-0036-4.
[20]	W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 15 (2002), 849-862. doi: 10.1088/0951-7715/15/3/320.
[21]	W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846. doi: 10.1017/S0143385703000543.
[22]	D. Kwietniak and P. Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Ergodic Theory Dynam. Systems, 32 (2012), 1661-1672. doi: 10.1017/S0143385711000599.
[23]	A. Leibman, Multiple recurrence theorem for nilpotent group actions, Geom. Funct. Anal., 4 (1994), 648-659. doi: 10.1007/BF01896657.
[24]	P. Walters, Some invariant $\sigma$-algebras for measure preserving transformations, Trans. Amer. Math. Soc., 163 (1972), 357-368. doi: 10.2307/1995727.

show all references

References:

[1]	D. Berend and V. Bergelson, Jointly ergodic measure-preserving transformations, Israel J. Math., 49 (1984), 307-314. doi: 10.1007/BF02760955.
[2]	V. Bergelson, Weakly mixing PET, Ergodic Theory Dynam. Systems, 7 (1987), 337-349. doi: 10.1017/S0143385700004090.
[3]	V. Bergelson, Combinatorial and Diophantine applications of ergodic theory, Handbook of Dynamical Systems, 1 (2006), 745-869. doi: 10.1016/S1874-575X(06)80037-8.
[4]	V. Bergelson and A. Leibman, Polynomial extensions of van der Waerden's and Szemerédi's theorems, J. Amer. Math. Soc., 9 (1996), 725-753. doi: 10.1090/S0894-0347-96-00194-4.
[5]	V. Bergelson and A. Leibman, Set-polynomials and polynomial extension of the Hales–Jewett theorem, Ann. of Math., 150 (1999), 33-75. doi: 10.2307/121097.
[6]	V. Bergelson, A. Leibman and Y. Son, Joint ergodicity along generalized linear functions, Ergodic Theory Dynam. Systems, 36 (2016), 2044-2075. doi: 10.1017/etds.2015.11.
[7]	V. Bergelson and R. McCutcheon, An ergodic IP polynomial Szemerédi theorem, Mem. Amer. Math. Soc., 146 (2000), 106pp. doi: 10.1090/memo/0695.
[8]	S. Donoso, A. Koutsogiannis and W. Sun, Seminorms for multiple averages along polynomials and applications to joint ergodicity, J. Anal. Math., arXiv: 1902.10237
[9]	N. Frantzikinakis and B. Kra, Polynomial averages converge to the product of integrals, Israel J. Math., 148 (2005), 267-276. doi: 10.1007/BF02775439.
[10]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N. J., 1981.
[11]	H. Furstenberg, IP-systems in Ergodic Theory, Contemp. Math., 26 (1984), 131-148. doi: 10.1090/conm/026/737395.
[12]	H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Anal. Math., 45 (1985), 117-168. doi: 10.1007/BF02792547.
[13]	H. Furstenberg and B. Weiss, The finite multipliers of infinite ergodic transformations, The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), 668 (1978), 127-132.
[14]	H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Anal. Math., 34 (1978), 61-85. doi: 10.1007/BF02790008.
[15]	E. Glasner, Topological ergodic decompositions and applications to products of powers of a minimal transformation, J. Anal. Math., 64 (1994), 241-262. doi: 10.1007/BF03008411.
[16]	E. Glasner, W. Huang, S. Shao, B. Weiss and X. Ye, Topological characteristic factors and nilsystems, arXiv: 2006.12385.
[17]	E. Glasner and B. Weiss, On the interplay between measurable and topological dynamics, Handbook of Dynamical Systems, 1 (2006), 597-648. doi: 10.1016/S1874-575X(06)80035-4.
[18]	N. Hindman, Finite sums from sequences within cells of a partition of ${\mathbb N}$, J. Combinatorial Theory Ser. A, 17 (1974), 1-11. doi: 10.1016/0097-3165(74)90023-5.
[19]	W. Huang, S. Shao and X. Ye, Topological correspondence of multiple ergodic averages of nilpotent group actions, J. Anal. Math., 138 (2019), 687-715. doi: 10.1007/s11854-019-0036-4.
[20]	W. Huang and X. Ye, An explicit scattering, non-weakly mixing example and weak disjointness, Nonlinearity, 15 (2002), 849-862. doi: 10.1088/0951-7715/15/3/320.
[21]	W. Huang and X. Ye, Topological complexity, return times and weak disjointness, Ergodic Theory Dynam. Systems, 24 (2004), 825-846. doi: 10.1017/S0143385703000543.
[22]	D. Kwietniak and P. Oprocha, On weak mixing, minimality and weak disjointness of all iterates, Ergodic Theory Dynam. Systems, 32 (2012), 1661-1672. doi: 10.1017/S0143385711000599.
[23]	A. Leibman, Multiple recurrence theorem for nilpotent group actions, Geom. Funct. Anal., 4 (1994), 648-659. doi: 10.1007/BF01896657.
[24]	P. Walters, Some invariant $\sigma$-algebras for measure preserving transformations, Trans. Amer. Math. Soc., 163 (1972), 357-368. doi: 10.2307/1995727.

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