Topological disjointness from entropy zero systems
[Disjonction topologique des systèmes d'entropie nulle]
Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 2, pp. 259-282.

Nous étudions les propriétés des systèmes topologiques dynamiques (X,T) qui sont disjoints de tout système minimal d’entropie nulle 0 . Contrairement au cas mesurable, il est connu que les K-systèmes topologiques constituent un sous-ensemble propre des systèmes disjoints de 0 . Nous montrons que (X,T) a une mesure invariante à support plein, et que si, de plus, (X,T) est transitif alors il est faiblement mélangeant. Nous construisons un système diagonal transitif avec un seul point minimal. Par conséquent, il existe un sous-ensemble grassement syndétique de  + , qui contient un sous-ensemble de  + , provenant d’un système minimal d’entropie positive, mais qui ne contienne aucun sous-ensemble de  + provenant d’un système minimal d’entropie nulle. D’autre part, nous étudions les propriétés des systèmes topologiques dynamiques (X,T) qui sont disjoints de classes plus larges de systèmes à entropie nulle.

The properties of topological dynamical systems (X,T) which are disjoint from all minimal systems of zero entropy, 0 , are investigated. Unlike the measurable case, it is known that topological K-systems make up a proper subset of the systems which are disjoint from 0 . We show that (X,T) has an invariant measure with full support, and if in addition (X,T) is transitive, then (X,T) is weakly mixing. A transitive diagonal system with only one minimal point is constructed. As a consequence, there exists a thickly syndetic subset of  + , which contains a subset of  + arising from a positive entropy minimal system, but does not contain any subset of  + arising from a zero entropy minimal system. Moreover we study the properties of topological dynamical systems (X,T) which are disjoint from larger classes of zero entropy systems.

DOI : 10.24033/bsmf.2534
Classification : 54H20
Keywords: disjointness, minimality, entropy, density
Mot clés : disjonction, minimalité, entropie, densité
@article{BSMF_2007__135_2_259_0,
     author = {Huang, Wen and Koh Park, Kyewon and Ye, Xiangdong},
     title = {Topological disjointness from entropy zero systems},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {259--282},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {135},
     number = {2},
     year = {2007},
     doi = {10.24033/bsmf.2534},
     mrnumber = {2430193},
     zbl = {1157.54015},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/bsmf.2534/}
}
TY  - JOUR
AU  - Huang, Wen
AU  - Koh Park, Kyewon
AU  - Ye, Xiangdong
TI  - Topological disjointness from entropy zero systems
JO  - Bulletin de la Société Mathématique de France
PY  - 2007
SP  - 259
EP  - 282
VL  - 135
IS  - 2
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/bsmf.2534/
DO  - 10.24033/bsmf.2534
LA  - en
ID  - BSMF_2007__135_2_259_0
ER  - 
%0 Journal Article
%A Huang, Wen
%A Koh Park, Kyewon
%A Ye, Xiangdong
%T Topological disjointness from entropy zero systems
%J Bulletin de la Société Mathématique de France
%D 2007
%P 259-282
%V 135
%N 2
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/bsmf.2534/
%R 10.24033/bsmf.2534
%G en
%F BSMF_2007__135_2_259_0
Huang, Wen; Koh Park, Kyewon; Ye, Xiangdong. Topological disjointness from entropy zero systems. Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 2, pp. 259-282. doi : 10.24033/bsmf.2534. http://archive.numdam.org/articles/10.24033/bsmf.2534/

[1] F. Blanchard - « Fully positive topological entropy and topological mixing », in Symbolic dynamics and its applications (New Haven, CT, 1991), Contemp. Math., vol. 135, Amer. Math. Soc., 1992, p. 95-105. | MR | Zbl

[2] -, « A disjointness theorem involving topological entropy », Bull. Soc. Math. France 121 (1993), p. 465-478. | Numdam | MR | Zbl

[3] F. Blanchard, B. Host & A. Maass - « Topological complexity », Ergodic Theory Dynam. Systems 20 (2000), p. 641-662. | MR | Zbl

[4] F. Blanchard & Y. Lacroix - « Zero entropy factors of topological flows », Proc. Amer. Math. Soc. 119 (1993), p. 985-992. | MR | Zbl

[5] H. Furstenberg - « Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation », Math. Systems Theory 1 (1967), p. 1-49. | MR | Zbl

[6] -, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, 1981, M. B. Porter Lectures. | MR | Zbl

[7] E. Glasner - « A simple characterization of the set of μ-entropy pairs and applications », Israel J. Math. 102 (1997), p. 13-27. | MR | Zbl

[8] E. Glasner & B. Weiss - « Quasi-factors of zero-entropy systems », J. Amer. Math. Soc. 8 (1995), p. 665-686. | MR | Zbl

[9] -, « Locally equicontinuous dynamical systems », Colloq. Math. 84/85 (2000), p. 345-361, Dedicated to the memory of Anzelm Iwanik. | MR

[10] W. H. He & Z. L. Zhou - « A topologically mixing system whose measure center is a singleton », Acta Math. Sinica (Chin. Ser.) 45 (2002), p. 929-934 (Chinese). | MR | Zbl

[11] W. Huang, S. Shao & X. Ye - « Mixing via sequence entropy », Comtemp. Math. 385 (2005), p. 101-122. | MR | Zbl

[12] W. Huang & X. Ye - « Generic eigenvalues, generic factors and weak disjointness », Preprint. | MR

[13] -, « Dynamical systems disjoint from any minimal system », Trans. Amer. Math. Soc. 357 (2005), p. 669-694. | MR | Zbl

[14] W. Parry - « Zero entropy of distal and related transformations », in Topological Dynamics (Symposium, Colorado State Univ., Ft. Collins, Colo., 1967), Benjamin, New York, 1968, p. 383-389. | MR | Zbl

[15] B. Song & X. Ye - « A minimal completely positive, non uniformly positive entropy example », to appear in J. Difference Equations and Applications. | MR | Zbl

[16] B. Weiss - « Topological transitivity and ergodic measures », Math. Systems Theory 5 (1971), p. 71-75. | MR | Zbl

Cité par Sources :