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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 |
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 $.
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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