Topological characteristic factors along cubes of minimal systems

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September 2019, 39(9): 5301-5317. doi: 10.3934/dcds.2019216

Topological characteristic factors along cubes of minimal systems

Fangzhou Cai and Song Shao ^,

Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

^* Corresponding author: Song Shao

Received September 2018 Revised February 2019 Published May 2019

Fund Project: This research is supported by NNSF of China (11571335, 11431012) and by “the Fundamental Research Funds for the Central Universities”.

In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal $ (d-1) $-step pro-nilfactor is the topological cubic characteristic factor of order $ d $.

Keywords: Topological characteristic factor, minimal system, pro-nilfactor.

Mathematics Subject Classification: Primary: 37B05; Secondary: 54H20.

Citation: Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5301-5317. doi: 10.3934/dcds.2019216

References:

[1]	E. Akin and E. Glasner, Topological ergodic decomposition and homogeneous flows, Topological Dynamics and Applications (Minneapolis, MN, 1995), 43–52, Contemp. Math., 215, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02929.
[2]	O. Antolin Camarena and B. Szegedy, Nilspaces, nilmanifolds and their morphisms, preprint, arXiv: 1009.3825.
[3]	P. Candela, Notes on nilspaces: Algebraic aspects, Discrete Anal., 2017 (2017), Paper No. 15, 59 pp.
[4]	P. Candela, Notes on compact nilspaces, Discrete Anal., 2017 (2017), Paper No. 16, 57 pp.
[5]	P. Dong, S. Donoso, A. Maass, S. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergod. Th. and Dynam. Sys., 33 (2013), 118-143. doi: 10.1017/S0143385711000861.
[6]	R. Ellis, S. Glasner and L. Shapiro, Proximal-Isometric Flows, Advances in Math, 17 (1975), 213-260. doi: 10.1016/0001-8708(75)90093-6.
[7]	H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math, 31 (1977), 204-256. doi: 10.1007/BF02813304.
[8]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981.
[9]	H. Furstenberg and B. Weiss, A mean ergodic theorem for $\frac{1}{N}\sum_\limits{n = 1}^Nf(T^nx)g(T^{n^2}x)$, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996,193–227.
[10]	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.
[11]	E. Glasner, $RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof, preprint, arXiv: 1402.3135.
[12]	E. Glasner, Y. Gutman and X. Ye, Higher order regionally proximal equivalence relations for general minimal group actions, Adv. Math., 333 (2018), 1004-1041. doi: 10.1016/j.aim.2018.05.023.
[13]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅰ., To appear in J. Analyse Math. doi: 10.1090/tran/7503.
[14]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅱ: Representation as nilmanifolds, Trans. Amer. Math. Soc., 371 (2019), 4951-4992. doi: 10.1090/tran/7503.
[15]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅲ: Inverse limit representations and topological dynamics, Submitted. http://arXiv.org/abs/1605.08950
[16]	B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397.
[17]	B. Host and B. Kra, Parallelepipeds, nilpotent groups and Gowers norms, Bull. Soc. Math. France, 136 (2008), 405-437. doi: 10.24033/bsmf.2561.
[18]	B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, Volume 236, American Mathematical Society, 2018.
[19]	B. Host, B. Kra and A. Maass, Nilsequences and a structure theory for topological dynamical systems, Advances in Mathematics, 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.
[20]	B. Host and A. Maass, Nilsystèmes d'ordre deux et parallélépipèdes, Bull. Soc. Math. France, 135 (2007), 367-405. doi: 10.24033/bsmf.2539.
[21]	W. Huang, S. Shao and X. D. Ye, Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems, preprint, 2017.
[22]	S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. in Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.
[23]	B. Szegedy, On higher order Fourier analysis, preprint, arXiv: 1203.2260.
[24]	J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.
[25]	T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97. doi: 10.1090/S0894-0347-06-00532-7.

show all references

References:

[1]	E. Akin and E. Glasner, Topological ergodic decomposition and homogeneous flows, Topological Dynamics and Applications (Minneapolis, MN, 1995), 43–52, Contemp. Math., 215, Amer. Math. Soc., Providence, RI, 1998. doi: 10.1090/conm/215/02929.
[2]	O. Antolin Camarena and B. Szegedy, Nilspaces, nilmanifolds and their morphisms, preprint, arXiv: 1009.3825.
[3]	P. Candela, Notes on nilspaces: Algebraic aspects, Discrete Anal., 2017 (2017), Paper No. 15, 59 pp.
[4]	P. Candela, Notes on compact nilspaces, Discrete Anal., 2017 (2017), Paper No. 16, 57 pp.
[5]	P. Dong, S. Donoso, A. Maass, S. Shao and X. Ye, Infinite-step nilsystems, independence and complexity, Ergod. Th. and Dynam. Sys., 33 (2013), 118-143. doi: 10.1017/S0143385711000861.
[6]	R. Ellis, S. Glasner and L. Shapiro, Proximal-Isometric Flows, Advances in Math, 17 (1975), 213-260. doi: 10.1016/0001-8708(75)90093-6.
[7]	H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math, 31 (1977), 204-256. doi: 10.1007/BF02813304.
[8]	H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981.
[9]	H. Furstenberg and B. Weiss, A mean ergodic theorem for $\frac{1}{N}\sum_\limits{n = 1}^Nf(T^nx)g(T^{n^2}x)$, Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State Univ. Math. Res. Inst. Publ., 5, de Gruyter, Berlin, 1996,193–227.
[10]	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.
[11]	E. Glasner, $RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof, preprint, arXiv: 1402.3135.
[12]	E. Glasner, Y. Gutman and X. Ye, Higher order regionally proximal equivalence relations for general minimal group actions, Adv. Math., 333 (2018), 1004-1041. doi: 10.1016/j.aim.2018.05.023.
[13]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅰ., To appear in J. Analyse Math. doi: 10.1090/tran/7503.
[14]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅱ: Representation as nilmanifolds, Trans. Amer. Math. Soc., 371 (2019), 4951-4992. doi: 10.1090/tran/7503.
[15]	Y. Gutman, F. Manners and P. Varjú, The structure theory of Nilspaces Ⅲ: Inverse limit representations and topological dynamics, Submitted. http://arXiv.org/abs/1605.08950
[16]	B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397.
[17]	B. Host and B. Kra, Parallelepipeds, nilpotent groups and Gowers norms, Bull. Soc. Math. France, 136 (2008), 405-437. doi: 10.24033/bsmf.2561.
[18]	B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, Volume 236, American Mathematical Society, 2018.
[19]	B. Host, B. Kra and A. Maass, Nilsequences and a structure theory for topological dynamical systems, Advances in Mathematics, 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.
[20]	B. Host and A. Maass, Nilsystèmes d'ordre deux et parallélépipèdes, Bull. Soc. Math. France, 135 (2007), 367-405. doi: 10.24033/bsmf.2539.
[21]	W. Huang, S. Shao and X. D. Ye, Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems, preprint, 2017.
[22]	S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. in Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.
[23]	B. Szegedy, On higher order Fourier analysis, preprint, arXiv: 1203.2260.
[24]	J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, 257. Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.
[25]	T. Ziegler, Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc., 20 (2007), 53-97. doi: 10.1090/S0894-0347-06-00532-7.

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