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

Topological characteristic factors along cubes of minimal systems

 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 $d$.

Citation: Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [2] O. Antolin Camarena and B. Szegedy, Nilspaces, nilmanifolds and their morphisms, preprint, arXiv: 1009.3825.Google Scholar [3] P. Candela, Notes on nilspaces: Algebraic aspects, Discrete Anal., 2017 (2017), Paper No. 15, 59 pp. Google Scholar [4] P. Candela, Notes on compact nilspaces, Discrete Anal., 2017 (2017), Paper No. 16, 57 pp. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981. Google Scholar [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. Google Scholar [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. Google Scholar [11] E. Glasner, $RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof, preprint, arXiv: 1402.3135.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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/1605.08950Google Scholar [16] B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397. Google Scholar [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. Google Scholar [18] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, Volume 236, American Mathematical Society, 2018. Google Scholar [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. Google Scholar [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. Google Scholar [21] W. Huang, S. Shao and X. D. Ye, Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems, preprint, 2017.Google Scholar [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. Google Scholar [23] B. Szegedy, On higher order Fourier analysis, preprint, arXiv: 1203.2260.Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [2] O. Antolin Camarena and B. Szegedy, Nilspaces, nilmanifolds and their morphisms, preprint, arXiv: 1009.3825.Google Scholar [3] P. Candela, Notes on nilspaces: Algebraic aspects, Discrete Anal., 2017 (2017), Paper No. 15, 59 pp. Google Scholar [4] P. Candela, Notes on compact nilspaces, Discrete Anal., 2017 (2017), Paper No. 16, 57 pp. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, M. B. Porter Lectures. Princeton University Press, Princeton, N.J., 1981. Google Scholar [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. Google Scholar [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. Google Scholar [11] E. Glasner, $RP^{[d]}$ is an equivalence relation: An enveloping semigroup proof, preprint, arXiv: 1402.3135.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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/1605.08950Google Scholar [16] B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math., 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397. Google Scholar [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. Google Scholar [18] B. Host and B. Kra, Nilpotent Structures in Ergodic Theory, Mathematical Surveys and Monographs, Volume 236, American Mathematical Society, 2018. Google Scholar [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. Google Scholar [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. Google Scholar [21] W. Huang, S. Shao and X. D. Ye, Regionally proximal relation of order $d$ along arithmetic progressions and nilsystems, preprint, 2017.Google Scholar [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. Google Scholar [23] B. Szegedy, On higher order Fourier analysis, preprint, arXiv: 1203.2260.Google Scholar [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. Google Scholar [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. Google Scholar
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