2015, 9: 365-405. doi: 10.3934/jmd.2015.9.365

Dynamical cubes and a criteria for systems having product extensions

1. 

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

2. 

Department of Mathematics, Northwestern University, 2033 Sheridan Road Evanston, IL 60208-2730, United States

Received  March 2015 Revised  November 2015 Published  December 2015

For minimal $\mathbb{Z}^{2}$-topological dynamical systems, we introduce a cube structure and a variation of the usual regional proximality relation for $\mathbb{Z}^2$ actions, which allow us to characterize product systems and their factors. We also introduce the concept of topological magic systems, which is the topological counterpart of measure theoretic magic systems introduced by Host in his study of multiple averages for commuting transformations. Roughly speaking, magic systems have less intricate dynamics, and we show that every minimal $\mathbb{Z}^2$ dynamical system has a magic extension. We give various applications of these structures, including the construction of some special factors in topological dynamics of $\mathbb{Z}^2$ actions and a computation of the automorphism group of the minimal Robinson tiling.
Citation: Sebastián Donoso, Wenbo Sun. Dynamical cubes and a criteria for systems having product extensions. Journal of Modern Dynamics, 2015, 9: 365-405. doi: 10.3934/jmd.2015.9.365
References:
[1]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.

[2]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Ann. Math. Studies, 53, Princeton University Press, 1963.

[3]

T. Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems, 30 (2010), 321-338. doi: 10.1017/S014338570900011X.

[4]

F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662. doi: 10.1017/S0143385700000341.

[5]

Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792. doi: 10.1017/S0143385710000258.

[6]

S. Donoso, Enveloping semigroups of systems of order d, Discrete Contin. Dyn. Sys., 34 (2014), 2729-2740. doi: 10.3934/dcds.2014.34.2729.

[7]

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

[8]

F. Gälher, A. Julien and J. Savinien, Combinatorics and topology of the Robinson tiling, C. R. Math. Acad. Sci. Paris, 350 (2012), 627-631. doi: 10.1016/j.crma.2012.06.007.

[9]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[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]

B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49. doi: 10.4064/sm195-1-3.

[12]

B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397-488. doi: 10.4007/annals.2005.161.397.

[13]

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.

[14]

W. Huang, S. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order, Memoirs of Amer. Math. Soc., 241, to appear. doi: 10.1090/memo/1143.

[15]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213. doi: 10.1017/S0143385704000215.

[16]

S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.

[17]

J. Olli, Endomorphisms of Sturmian systems and the discrete chair substitution tiling system, Discrete Contin. Dyn. Syst., 33 (2013), 4173-4186. doi: 10.3934/dcds.2013.33.4173.

[18]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7.

[19]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Second edition, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[20]

C. Radin, Miles of Tiles, Student Mathematical Library, 1, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/stml/001.

[21]

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.

[22]

T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.

[23]

H. Towsner, Convergence of diagonal ergodic averages, Ergodic Theory Dynam. Systems, 29 (2009), 1309-1326. doi: 10.1017/S0143385708000722.

[24]

S. Tu and X. Ye, Dynamical parallelepipeds in minimal systems, J. Dynam. Differential Equations, 25 (2013), 765-776. doi: 10.1007/s10884-013-9313-6.

show all references

References:
[1]

J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.

[2]

L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Ann. Math. Studies, 53, Princeton University Press, 1963.

[3]

T. Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems, 30 (2010), 321-338. doi: 10.1017/S014338570900011X.

[4]

F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662. doi: 10.1017/S0143385700000341.

[5]

Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792. doi: 10.1017/S0143385710000258.

[6]

S. Donoso, Enveloping semigroups of systems of order d, Discrete Contin. Dyn. Sys., 34 (2014), 2729-2740. doi: 10.3934/dcds.2014.34.2729.

[7]

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

[8]

F. Gälher, A. Julien and J. Savinien, Combinatorics and topology of the Robinson tiling, C. R. Math. Acad. Sci. Paris, 350 (2012), 627-631. doi: 10.1016/j.crma.2012.06.007.

[9]

E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.

[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]

B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49. doi: 10.4064/sm195-1-3.

[12]

B. Host and B. Kra, Nonconventional averages and nilmanifolds, Ann. of Math. (2), 161 (2005), 397-488. doi: 10.4007/annals.2005.161.397.

[13]

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.

[14]

W. Huang, S. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order, Memoirs of Amer. Math. Soc., 241, to appear. doi: 10.1090/memo/1143.

[15]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213. doi: 10.1017/S0143385704000215.

[16]

S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.

[17]

J. Olli, Endomorphisms of Sturmian systems and the discrete chair substitution tiling system, Discrete Contin. Dyn. Syst., 33 (2013), 4173-4186. doi: 10.3934/dcds.2013.33.4173.

[18]

K. E. Petersen, Disjointness and weak mixing of minimal sets, Proc. Amer. Math. Soc., 24 (1970), 278-280. doi: 10.1090/S0002-9939-1970-0250283-7.

[19]

M. Queffélec, Substitution Dynamical Systems-Spectral Analysis, Second edition, Lecture Notes in Mathematics, 1294, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-11212-6.

[20]

C. Radin, Miles of Tiles, Student Mathematical Library, 1, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/stml/001.

[21]

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.

[22]

T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.

[23]

H. Towsner, Convergence of diagonal ergodic averages, Ergodic Theory Dynam. Systems, 29 (2009), 1309-1326. doi: 10.1017/S0143385708000722.

[24]

S. Tu and X. Ye, Dynamical parallelepipeds in minimal systems, J. Dynam. Differential Equations, 25 (2013), 765-776. doi: 10.1007/s10884-013-9313-6.

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