American Institute of Mathematical Sciences

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.  Google Scholar [2] L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Ann. Math. Studies, 53, Princeton University Press, 1963. Google Scholar [3] T. Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems, 30 (2010), 321-338. doi: 10.1017/S014338570900011X.  Google Scholar [4] F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662. doi: 10.1017/S0143385700000341.  Google Scholar [5] Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792. doi: 10.1017/S0143385710000258.  Google Scholar [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.  Google Scholar [7] R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.  Google Scholar [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.  Google Scholar [9] E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar [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.  Google Scholar [11] B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49. doi: 10.4064/sm195-1-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Huang, S. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order,, Memoirs of Amer. Math. Soc., 241 ().  doi: 10.1090/memo/1143.  Google Scholar [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.  Google Scholar [16] S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] C. Radin, Miles of Tiles, Student Mathematical Library, 1, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/stml/001.  Google Scholar [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.  Google Scholar [22] T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.  Google Scholar [23] H. Towsner, Convergence of diagonal ergodic averages, Ergodic Theory Dynam. Systems, 29 (2009), 1309-1326. doi: 10.1017/S0143385708000722.  Google Scholar [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.  Google Scholar

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References:
 [1] J. Auslander, Minimal Flows and Their Extensions, North-Holland Mathematics Studies, 153, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar [2] L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces, Ann. Math. Studies, 53, Princeton University Press, 1963. Google Scholar [3] T. Austin, On the norm convergence of non-conventional ergodic averages, Ergodic Theory Dynam. Systems, 30 (2010), 321-338. doi: 10.1017/S014338570900011X.  Google Scholar [4] F. Blanchard, B. Host and A. Maass, Topological complexity, Ergodic Theory Dynam. Systems, 20 (2000), 641-662. doi: 10.1017/S0143385700000341.  Google Scholar [5] Q. Chu, Multiple recurrence for two commuting transformations, Ergodic Theory Dynam. Systems, 31 (2011), 771-792. doi: 10.1017/S0143385710000258.  Google Scholar [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.  Google Scholar [7] R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.  Google Scholar [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.  Google Scholar [9] E. Glasner, Ergodic Theory via Joinings, Mathematical Surveys and Monographs, 101, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/surv/101.  Google Scholar [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.  Google Scholar [11] B. Host, Ergodic seminorms for commuting transformations and applications, Studia Math., 195 (2009), 31-49. doi: 10.4064/sm195-1-3.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Huang, S. Shao and X. Ye, Nil Bohr-sets and almost automorphy of higher order,, Memoirs of Amer. Math. Soc., 241 ().  doi: 10.1090/memo/1143.  Google Scholar [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.  Google Scholar [16] S. Mozes, Tilings, substitution systems and dynamical systems generated by them, J. Anal. Math., 53 (1989), 139-186. doi: 10.1007/BF02793412.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] C. Radin, Miles of Tiles, Student Mathematical Library, 1, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/stml/001.  Google Scholar [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.  Google Scholar [22] T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.  Google Scholar [23] H. Towsner, Convergence of diagonal ergodic averages, Ergodic Theory Dynam. Systems, 29 (2009), 1309-1326. doi: 10.1017/S0143385708000722.  Google Scholar [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.  Google Scholar
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