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

Enveloping semigroups of systems of order d

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.
Citation: Sebastián Donoso. Enveloping semigroups of systems of order d. Discrete and 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

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