Directional dynamical cubes for minimal Zd-systems

37B05 (Secondary); 54H20 (Primary); commuting transformations 2010 Mathematics Subject Classification; cubespaces; directional cubes; minimal systems; topological dynamics

Abstract :

[en] We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal -system. We study the structural properties of systems that satisfy the so-called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a -system that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal -systems that enjoy the unique closing parallelepiped property and provide explicit examples.

Disciplines :

Mathematics

Author, co-author :

Cabezas Aros, Christopher ; Université de Liège - ULiège > Mathematics ; Departamento de Ingenieria Matematica and Centro de Modelamiento Matematico, Universidad de Chile AND UMI-CNRS 2807, Santiago, Chile

Donoso, Sebastian ; Universidad de Chile > Departamento de Ingeniería Matemática

Maass, Alejandro ; Departamento de Ingenieria Matematica and Centro de Modelamiento Matematico, Universidad de Chile AND UMI-CNRS 2807, Santiago, Chile

Language :

English

Title :

Directional dynamical cubes for minimal Zd-systems

Publication date :

December 2020

Journal title :

Ergodic Theory and Dynamical Systems

ISSN :

0143-3857

eISSN :

1469-4417

Publisher :

Cambridge University Press

Volume :

Issue :

Pages :

3257 - 3295

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0143385719000336

Funding text :

Acknowledgement. This work was funded by Proyecto/Grant PIA AFB-170001 and Math-AmSud DCS 17 - MATH 01. The second author is also supported by Fondecyt Iniciación Grant 11160061. We thank Wenbo Sun for many helpful discussions. We also thank the anonymous referee for valuable remarks.

Available on ORBi :

since 15 January 2024

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Bibliography

O. Antolín Camarena and B. Szegedy. Nilspaces, nilmanifolds and their morphisms. Preprint, 2010, arXiv:1009. 3825.
J. Auslander. Minimal flows and their extensions. Notas de Matemática [Mathematical Notes], Volume 122 (North-Holland Mathematics Studies, 153). North-Holland, Amsterdam, 1988.
T. Austin. On the norm convergence of non-conventional ergodic averages. Ergod. Th. & Dynam. Sys. 30(2) (2010), 321-338.
T. Austin. Multiple recurrence and the structure of probability-preserving systems. PhD Thesis, University of California, Los Angeles, 2010.
C. Cabezas. Cubos dinámicos direccionales para Zd-sistemas minimales. Master's Thesis, Universidad de Chile, 2018.
P. Candela. Notes on nilspaces: algebraic aspects. Discrete Anal. (2017), Paper No. 15.
P. Candela. Notes on compact nilspaces. Discrete Anal. (2017), Paper No. 16.
S. Donoso andW. Sun. Dynamical cubes and a criteria for systems having product extensions. J. Mod. Dyn. 9 (2015), 365-405.
S. Donoso and W. Sun. A pointwise cubic average for two commuting transformations. Israel J. Math. 216(2) (2016), 657-678.
S. Donoso and W. Sun. Pointwise multiple averages for systems with two commuting transformations. Ergod. Th. & Dynam. Sys. 38(6) (2018), 2132-2157.
S. Donoso and W. Sun. Pointwise convergence of some multiple ergodic averages. Adv. Math. 330 (2018), 946-996.
E. Glasner. Topological ergodic decompositions and applications to products of powers of a minimal transformation. J. Anal. Math. 64 (1994), 241-262.
E. Glasner, Y. Gutman and X. Ye. Higher order regionally proximal equivalence relations for general minimal group actions. Adv. Math. 333 (2018), 1004-1041.
Y. Gutman, F. Manners and P. Varjú. The structure theory of nilspaces III: Inverse limit representations and topological dynamics. Preprint, 2016, arXiv:1605. 08950.
Y. Gutman, F. Manners and P. Varjú. The structure theory of nilspaces I. J. Anal. Math. (2018), in press.
Y. Gutman, F. Manners and P. Varjú. The structure theory of nilspaces II: Representation as nilmanifolds. Trans. Amer. Math. Soc. 371 (2019), 4951-4992.
B. Host. Ergodic seminorms for commuting transformations and applications. Studia Math. 195(1) (2009), 31-49.
B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161(1) (2005), 397-488.
B. Host, B. Kra and A. Maass. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math. 224(1) (2010), 103-129.
W. Huang, S. Shao and X. Ye. Nil Bohr-sets and almost automorphy of higher order. Mem. Amer. Math. Soc. 241(1143) (2016).
A. Leibman. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25(1) (2005), 201-213.
W. Parry. Ergodic properties of affine transformations and flows on nilmanifolds. Amer. J. Math. 91 (1969), 757-771.
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(3-4) (2012), 1786-1817.
T. Tao. Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys. 28(2) (2008), 657-688.
S. Tu and X. Ye. Dynamical parallelepipeds in minimal systems. J. Dynam. Differential Equations 25(3) (2013), 765-776.