Enveloping semigroups of systems of order d

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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

Abstract
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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 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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