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July  2014, 34(7): 2729-2740. doi: 10.3934/dcds.2014.34.2729

Enveloping semigroups of systems of order d

Sebastián Donoso 1,

1. 

Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile

Received  July 2013 Revised  October 2013 Published  December 2013

In this paper we study the Ellis semigroup of a $d$-step nilsystem and the inverse limit of such systems. By using the machinery of cubes developed by Host, Kra and Maass, we prove that such a system has a $d$-step topologically nilpotent enveloping semigroup. In the case $d=2$, we prove that these notions are equivalent, extending a previous result by Glasner.
Keywords: minimal systems., nilsystems, regionally proximal relation, enveloping semigroup, Topological dynamics.
Mathematics Subject Classification: Primary: 37B05, 37B9.
Citation: Sebastián Donoso. Enveloping semigroups of systems of order d. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2729-2740. doi: 10.3934/dcds.2014.34.2729
References:
[1]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153, Notas de Matemática [Mathematical Notes], 122, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[2]

H. Becker and A. S. Kechris, The Descriptive set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, 232, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511735264.  Google Scholar

[3]

R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.  Google Scholar

[4]

E. Glasner, Minimal nil-transformations of class two, Israel J. Math., 81 (1993), 31-51. doi: 10.1007/BF02761296.  Google Scholar

[5]

E. Glasner, Enveloping semigroups in topological dynamics, Topology Appl., 154 (2007), 2344-2363. doi: 10.1016/j.topol.2007.03.009.  Google Scholar

[6]

B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2), 171 (2010), 1753-1850. doi: 10.4007/annals.2010.171.1753.  Google Scholar

[7]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397.  Google Scholar

[8]

B. Host, B. Kra and A. Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math., 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.  Google Scholar

[9]

B. Host and A. Maass, Nilsystèmes d'ordre deux et parallélépipèdes, (French) [Two step nilsystems and parallelepipeds], Bull. Soc. Math. France, 135 (2007), 367-405.  Google Scholar

[10]

A. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk SSSR. Ser Mat., 13 (1949), 9-32.  Google Scholar

[11]

R. Pikuła, Enveloping semigroups of unipotent affine transformations of the torus, Ergodic Theory Dynam. Systems, 30 (2010), 1543-1559. doi: 10.1017/S0143385709000261.  Google Scholar

[12]

R. J. Sacker and G. R. Sell, Finite extensions of minimal transformation groups, Trans. Amer. Math. Soc., 190 (1974), 325-334. doi: 10.1090/S0002-9947-1974-0350715-8.  Google Scholar

[13]

S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.  Google Scholar

show all references

References:
[1]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, 153, Notas de Matemática [Mathematical Notes], 122, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[2]

H. Becker and A. S. Kechris, The Descriptive set Theory of Polish Group Actions, London Mathematical Society Lecture Note Series, 232, Cambridge Univ. Press, Cambridge, 1996. doi: 10.1017/CBO9780511735264.  Google Scholar

[3]

R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.  Google Scholar

[4]

E. Glasner, Minimal nil-transformations of class two, Israel J. Math., 81 (1993), 31-51. doi: 10.1007/BF02761296.  Google Scholar

[5]

E. Glasner, Enveloping semigroups in topological dynamics, Topology Appl., 154 (2007), 2344-2363. doi: 10.1016/j.topol.2007.03.009.  Google Scholar

[6]

B. Green and T. Tao, Linear equations in primes, Ann. of Math. (2), 171 (2010), 1753-1850. doi: 10.4007/annals.2010.171.1753.  Google Scholar

[7]

B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 398-488. doi: 10.4007/annals.2005.161.397.  Google Scholar

[8]

B. Host, B. Kra and A. Maass, Nilsequences and a structure theorem for topological dynamical systems, Adv. Math., 224 (2010), 103-129. doi: 10.1016/j.aim.2009.11.009.  Google Scholar

[9]

B. Host and A. Maass, Nilsystèmes d'ordre deux et parallélépipèdes, (French) [Two step nilsystems and parallelepipeds], Bull. Soc. Math. France, 135 (2007), 367-405.  Google Scholar

[10]

A. Mal'cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk SSSR. Ser Mat., 13 (1949), 9-32.  Google Scholar

[11]

R. Pikuła, Enveloping semigroups of unipotent affine transformations of the torus, Ergodic Theory Dynam. Systems, 30 (2010), 1543-1559. doi: 10.1017/S0143385709000261.  Google Scholar

[12]

R. J. Sacker and G. R. Sell, Finite extensions of minimal transformation groups, Trans. Amer. Math. Soc., 190 (1974), 325-334. doi: 10.1090/S0002-9947-1974-0350715-8.  Google Scholar

[13]

S. Shao and X. Ye, Regionally proximal relation of order $d$ is an equivalence one for minimal systems and a combinatorial consequence, Adv. Math., 231 (2012), 1786-1817. doi: 10.1016/j.aim.2012.07.012.  Google Scholar

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