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.
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.
Keywords: Product recurrence, weak mixing, minimal system, disjointness
Mot clés : récurrence produit, mélange faible, système minimal, disjonction
Oprocha, Piotr 1, 2
@article{AIF_2010__60_4_1233_0, author = {Oprocha, Piotr}, title = {Weak mixing and product recurrence}, journal = {Annales de l'Institut Fourier}, pages = {1233--1257}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {60}, number = {4}, year = {2010}, doi = {10.5802/aif.2553}, mrnumber = {2722240}, zbl = {1203.37026}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2553/} }
TY - JOUR AU - Oprocha, Piotr TI - Weak mixing and product recurrence JO - Annales de l'Institut Fourier PY - 2010 SP - 1233 EP - 1257 VL - 60 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2553/ DO - 10.5802/aif.2553 LA - en ID - AIF_2010__60_4_1233_0 ER -
%0 Journal Article %A Oprocha, Piotr %T Weak mixing and product recurrence %J Annales de l'Institut Fourier %D 2010 %P 1233-1257 %V 60 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2553/ %R 10.5802/aif.2553 %G en %F AIF_2010__60_4_1233_0
Oprocha, Piotr. Weak mixing and product recurrence. Annales de l'Institut Fourier, Volume 60 (2010) no. 4, pp. 1233-1257. doi : 10.5802/aif.2553. https://aif.centre-mersenne.org/articles/10.5802/aif.2553/
[1] Product recurrence and distal points, Trans. Amer. Math. Soc., Volume 343 (1994) no. 1, pp. 221-232 | DOI | MR | Zbl
[2] Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, Volume 17 (1997) no. 3, pp. 505-529 | DOI | MR | Zbl
[3] Topological mapping properties defined by digraphs, Discrete Contin. Dynam. Systems, Volume 5 (1999) no. 1, pp. 83-92 | DOI | MR | Zbl
[4] Entropy sets, weakly mixing sets and entropy capacity, Discrete Contin. Dyn. Syst., Volume 20 (2008) no. 2, pp. 275-311 | MR | Zbl
[5] 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] Semicocycle extensions and the stroboscopic property, Topology Appl., Volume 153 (2005) no. 1, pp. 97-106 | DOI | MR | Zbl
[7] 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] Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981 (M. B. Porter Lectures) | MR | Zbl
[9] Chaos on hyperspaces, Nonlinear Anal., Volume 71 (2009) no. 1-2, pp. 1-8 | DOI | MR | Zbl
[10] Recurrence in pairs, Ergodic Theory Dynam. Systems, Volume 28 (2008) no. 4, pp. 1135-1143 | DOI | MR | Zbl
[11] Dynamical systems disjoint from any minimal system, Trans. Amer. Math. Soc., Volume 357 (2005) no. 2, pp. 669-694 | DOI | MR | Zbl
[12] On density of periodic points for induced hyperspace maps, Top. Proc., Volume 35 (2010), pp. 281-290 | MR | Zbl
[13] Spectral decomposition theorem for non-hyperbolic maps, Dyn. Syst., Volume 23 (2008) no. 3, pp. 299-307 | DOI | MR | Zbl
[14] Differentiable dynamical systems, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 747-817 | DOI | MR | Zbl
Cited by Sources: