# 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, (1988). Google Scholar [2] L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces,, Ann. Math. Studies, (1963). Google Scholar [3] T. Austin, On the norm convergence of non-conventional ergodic averages,, Ergodic Theory Dynam. Systems, 30 (2010), 321. doi: 10.1017/S014338570900011X. Google Scholar [4] F. Blanchard, B. Host and A. Maass, Topological complexity,, Ergodic Theory Dynam. Systems, 20 (2000), 641. doi: 10.1017/S0143385700000341. Google Scholar [5] Q. Chu, Multiple recurrence for two commuting transformations,, Ergodic Theory Dynam. Systems, 31 (2011), 771. doi: 10.1017/S0143385710000258. Google Scholar [6] S. Donoso, Enveloping semigroups of systems of order d,, Discrete Contin. Dyn. Sys., 34 (2014), 2729. doi: 10.3934/dcds.2014.34.2729. Google Scholar [7] R. Ellis, Lectures on Topological Dynamics,, W. A. Benjamin, (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. doi: 10.1016/j.crma.2012.06.007. Google Scholar [9] E. Glasner, Ergodic Theory via Joinings,, Mathematical Surveys and Monographs, (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. doi: 10.1007/BF03008411. Google Scholar [11] B. Host, Ergodic seminorms for commuting transformations and applications,, Studia Math., 195 (2009), 31. 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. 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. 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. doi: 10.1017/S0143385704000215. Google Scholar [16] S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139. 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. 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. doi: 10.1090/S0002-9939-1970-0250283-7. Google Scholar [19] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis,, Second edition, (1294). doi: 10.1007/978-3-642-11212-6. Google Scholar [20] C. Radin, Miles of Tiles,, Student Mathematical Library, (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. 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. doi: 10.1017/S0143385708000011. Google Scholar [23] H. Towsner, Convergence of diagonal ergodic averages,, Ergodic Theory Dynam. Systems, 29 (2009), 1309. doi: 10.1017/S0143385708000722. Google Scholar [24] S. Tu and X. Ye, Dynamical parallelepipeds in minimal systems,, J. Dynam. Differential Equations, 25 (2013), 765. doi: 10.1007/s10884-013-9313-6. Google Scholar

show all references

##### References:
 [1] J. Auslander, Minimal Flows and Their Extensions,, North-Holland Mathematics Studies, (1988). Google Scholar [2] L. Auslander, L. Green and F. Hahn, Flows on Homogeneous Spaces,, Ann. Math. Studies, (1963). Google Scholar [3] T. Austin, On the norm convergence of non-conventional ergodic averages,, Ergodic Theory Dynam. Systems, 30 (2010), 321. doi: 10.1017/S014338570900011X. Google Scholar [4] F. Blanchard, B. Host and A. Maass, Topological complexity,, Ergodic Theory Dynam. Systems, 20 (2000), 641. doi: 10.1017/S0143385700000341. Google Scholar [5] Q. Chu, Multiple recurrence for two commuting transformations,, Ergodic Theory Dynam. Systems, 31 (2011), 771. doi: 10.1017/S0143385710000258. Google Scholar [6] S. Donoso, Enveloping semigroups of systems of order d,, Discrete Contin. Dyn. Sys., 34 (2014), 2729. doi: 10.3934/dcds.2014.34.2729. Google Scholar [7] R. Ellis, Lectures on Topological Dynamics,, W. A. Benjamin, (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. doi: 10.1016/j.crma.2012.06.007. Google Scholar [9] E. Glasner, Ergodic Theory via Joinings,, Mathematical Surveys and Monographs, (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. doi: 10.1007/BF03008411. Google Scholar [11] B. Host, Ergodic seminorms for commuting transformations and applications,, Studia Math., 195 (2009), 31. 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. 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. 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. doi: 10.1017/S0143385704000215. Google Scholar [16] S. Mozes, Tilings, substitution systems and dynamical systems generated by them,, J. Anal. Math., 53 (1989), 139. 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. 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. doi: 10.1090/S0002-9939-1970-0250283-7. Google Scholar [19] M. Queffélec, Substitution Dynamical Systems-Spectral Analysis,, Second edition, (1294). doi: 10.1007/978-3-642-11212-6. Google Scholar [20] C. Radin, Miles of Tiles,, Student Mathematical Library, (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. 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. doi: 10.1017/S0143385708000011. Google Scholar [23] H. Towsner, Convergence of diagonal ergodic averages,, Ergodic Theory Dynam. Systems, 29 (2009), 1309. doi: 10.1017/S0143385708000722. Google Scholar [24] S. Tu and X. Ye, Dynamical parallelepipeds in minimal systems,, J. Dynam. Differential Equations, 25 (2013), 765. doi: 10.1007/s10884-013-9313-6. Google Scholar
 [1] Fangzhou Cai, Song Shao. Topological characteristic factors along cubes of minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5301-5317. doi: 10.3934/dcds.2019216 [2] Patrick Bonckaert, Timoteo Carletti, Ernest Fontich. On dynamical systems close to a product of $m$ rotations. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 349-366. doi: 10.3934/dcds.2009.24.349 [3] Jean-Luc Chabert, Ai-Hua Fan, Youssef Fares. Minimal dynamical systems on a discrete valuation domain. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 777-795. doi: 10.3934/dcds.2009.25.777 [4] Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785 [5] H. M. Hastings, S. Silberger, M. T. Weiss, Y. Wu. A twisted tensor product on symbolic dynamical systems and the Ashley's problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 549-558. doi: 10.3934/dcds.2003.9.549 [6] Patrik Nystedt, Johan Öinert. Simple skew category algebras associated with minimal partially defined dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4157-4171. doi: 10.3934/dcds.2013.33.4157 [7] Younghwan Son. Substitutions, tiling dynamical systems and minimal self-joinings. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4855-4874. doi: 10.3934/dcds.2014.34.4855 [8] Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258 [9] Stefano Luzzatto, Marks Ruziboev. Young towers for product systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1465-1491. doi: 10.3934/dcds.2016.36.1465 [10] Nir Avni, Benjamin Weiss. Generating product systems. Journal of Modern Dynamics, 2010, 4 (2) : 257-270. doi: 10.3934/jmd.2010.4.257 [11] P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 [12] Péter Koltai, Alexander Volf. Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times. Journal of Computational Dynamics, 2014, 1 (2) : 339-356. doi: 10.3934/jcd.2014.1.339 [13] Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i [14] Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116 [15] Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017 [16] El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449 [17] Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355 [18] Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447 [19] John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361 [20] Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935

2018 Impact Factor: 0.295

## Metrics

• PDF downloads (5)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]