no. 4
Weak mixing and product recurrence
[Mélange faible et récurrence produit]
Oprocha, Piotr1, 2
1 Universidad de Murcia Departamento de Matemáticas Campus de Espinardo 30100 Murcia (Spain)
2 AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30 30-059 Kraków (Poland)
Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1233-1257.

Dans cet article nous étudions la structure de l’ensemble des points faiblement produit-récurrents. Nous donnons entre autres des conditions suffisantes (en rapport avec le mélange topologique faible) qui impliquent que l’ensemble des points faiblement produit-récurrents est résiduel. De plus, nous obtenons certains résultats nouveaux concernant la classe des systèmes disjoints de tous les systèmes minimaux.

In this article we study the structure of the set of weakly product recurrent points. Among others, we provide necessary conditions (related to topological weak mixing) which imply that the set of weakly product recurrent points is residual. Additionally, some new results about the class of systems disjoint from every minimal system are obtained.

DOI : 10.5802/aif.2553
Classification : 37B20, 37B05
Keywords: Product recurrence, weak mixing, minimal system, disjointness
Mot clés : récurrence produit, mélange faible, système minimal, disjonction
Oprocha, Piotr 1, 2

1 Universidad de Murcia Departamento de Matemáticas Campus de Espinardo 30100 Murcia (Spain)
2 AGH University of Science and Technology Faculty of Applied Mathematics al. Mickiewicza 30 30-059 Kraków (Poland) 
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Oprocha, Piotr. Weak mixing and product recurrence. Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1233-1257. doi : 10.5802/aif.2553. http://www.numdam.org/articles/10.5802/aif.2553/

[1] Auslander, J.; Furstenberg, H. Product recurrence and distal points, Trans. Amer. Math. Soc., Volume 343 (1994) no. 1, pp. 221-232 | DOI | MR | Zbl

[2] Banks, John Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, Volume 17 (1997) no. 3, pp. 505-529 | DOI | MR | Zbl

[3] Banks, John Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems, Volume 5 (1999) no. 1, pp. 83-92 | DOI | MR | Zbl

[4] Blanchard, François; Huang, Wen Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., Volume 20 (2008) no. 2, pp. 275-311 | MR | Zbl

[5] Downarowicz, Tomasz Survey of odometers and Toeplitz flows, Algebraic and topological dynamics (Contemp. Math.), Volume 385, Amer. Math. Soc., Providence, RI, 2005, pp. 7-37 | MR | Zbl

[6] Downarowicz, Tomasz; Serafin, Jacek Semicocycle extensions and the stroboscopic property, Topology Appl., Volume 153 (2005) no. 1, pp. 97-106 | DOI | MR | Zbl

[7] Furstenberg, H. Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, Volume 1 (1967), pp. 1-49 | DOI | MR | Zbl

[8] Furstenberg, H. Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981 (M. B. Porter Lectures) | MR | Zbl

[9] Guirao, Juan Luis García; Kwietniak, Dominik; Lampart, Marek; Oprocha, Piotr; Peris, Alfredo Chaos on hyperspaces, Nonlinear Anal., Volume 71 (2009) no. 1-2, pp. 1-8 | DOI | MR | Zbl

[10] Haddad, Kamel; Ott, William Recurrence in pairs, Ergodic Theory Dynam. Systems, Volume 28 (2008) no. 4, pp. 1135-1143 | DOI | MR | Zbl

[11] Huang, Wen; Ye, Xiangdong Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., Volume 357 (2005) no. 2, pp. 669-694 | DOI | MR | Zbl

[12] Méndez, Héctor On density of periodic points for induced hyperspace maps, Top. Proc., Volume 35 (2010), pp. 281-290 | MR | Zbl

[13] Oprocha, Piotr Spectral decomposition theorem for non-hyperbolic maps, Dyn. Syst., Volume 23 (2008) no. 3, pp. 299-307 | DOI | MR | Zbl

[14] Smale, S. Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 | DOI | MR | Zbl

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