no. 4
Sequence entropy pairs and complexity pairs for a measure
[Paires d'entropie séquentielle et paires de complexité pour une mesure]
Huang, Wen1 ; Maass, Alejandro ; Ye, Xiangdong
1 University of Science and Technology of China, Department of Mathematics, Hefei, Anhui, 230026 P.R. (Chine), Universidad de Chile, Departamento de Ingenier\'ia Matemática and Centro de Modelamiento Matemático, Casilla 170/3 correo 3, Santiago (Chili), University of Science and Technology of China, Department of Mathematics, Hefei, Anhui, 230026 P.R. (Chine)
Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1005-1028.

Dans cet article, nous étudions des facteurs topologiques entre le facteur de Kronecker et le facteur équicontinu maximal d’un système dynamique. Nous introduisons la notion de n-tuple d’entropie séquentielle pour une mesure et nous prouvons que l’ensemble n- tuple d’entropie sequentielle pour une mesure est contenu dans l’ensemble de n-tuple d’entropie séquentielle topologique [H-Y]. La réciproque est fausse. Aussi en suivant les idées dans [BHM], nous introduisons une notion faible et une notion forte de paire de complexité pour une mesure. Nous prouvons que la notion forte est strictement contenue entre la notion de paire d’entropie et de paire de complexité topologique.

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy n-tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a measure. We prove that in general the strongest notion is strictly contained in between sequence entropy pairs and topological complexity pairs.

DOI : 10.5802/aif.2041
Classification : 54H20
Keywords: sequential entropy, complexity
Mot clés : entropie séquentielle, complexité

Huang, Wen 1 ; Maass, Alejandro  ; Ye, Xiangdong 

1 University of Science and Technology of China, Department of Mathematics, Hefei, Anhui, 230026 P.R. (Chine), Universidad de Chile, Departamento de Ingenier\'ia Matemática and Centro de Modelamiento Matemático, Casilla 170/3 correo 3, Santiago (Chili), University of Science and Technology of China, Department of Mathematics, Hefei, Anhui, 230026 P.R. (Chine) 
@article{AIF_2004__54_4_1005_0,
     author = {Huang, Wen and Maass, Alejandro and Ye, Xiangdong},
     title = {Sequence entropy pairs and complexity pairs for a measure},
     journal = {Annales de l'Institut Fourier},
     pages = {1005--1028},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {4},
     year = {2004},
     doi = {10.5802/aif.2041},
     zbl = {1083.37006},
     mrnumber = {2111019},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2041/}
}
TY  - JOUR
AU  - Huang, Wen
AU  - Maass, Alejandro
AU  - Ye, Xiangdong
TI  - Sequence entropy pairs and complexity pairs for a measure
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 1005
EP  - 1028
VL  - 54
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2041/
DO  - 10.5802/aif.2041
LA  - en
ID  - AIF_2004__54_4_1005_0
ER  - 
%0 Journal Article
%A Huang, Wen
%A Maass, Alejandro
%A Ye, Xiangdong
%T Sequence entropy pairs and complexity pairs for a measure
%J Annales de l'Institut Fourier
%D 2004
%P 1005-1028
%V 54
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2041/
%R 10.5802/aif.2041
%G en
%F AIF_2004__54_4_1005_0
Huang, Wen; Maass, Alejandro; Ye, Xiangdong. Sequence entropy pairs and complexity pairs for a measure. Annales de l'Institut Fourier, Tome 54 (2004) no. 4, pp. 1005-1028. doi : 10.5802/aif.2041. https://aif.centre-mersenne.org/articles/10.5802/aif.2041/

[B-R] F. Blanchard; B. Host; A. Maass; S. Martínez; D. Rudolph Entropy pairs for a measure, Ergod. Th. and Dynam. Sys, Volume 15 (1995), pp. 621-632 | MR | Zbl

[B1] F. Blanchard Fully positive topological entropy and topological mixing, Symbolic dynamics and its applications, AMS Contemporary Mathematics, Volume 135 (1992), pp. 95-105 | MR | Zbl

[B2] F. Blanchard A disjointness theorem involving topological entropy, Bull. de la Soc. Math. de France, Volume 121 (1993), pp. 465-478 | Numdam | MR | Zbl

[Be] V. Bergelson Ergodic Ramsey theory -- an update, Ergodic theory of ℤ d actions (Warwick, 1993-1994) (London Math. Soc. Lecture Notes Ser.), Volume 228 (1996), pp. 1-61 | Zbl

[BGH] F. Blanchard; E. Glasner; B. Host Variations on the variational principle, Ergod. Th. and Dynam. Sys, Volume 17 (1997), pp. 29-53 | MR | Zbl

[BGKM] F. Blanchard; E. Glasner; S. Kolyada; A. Maass On Li-Yorke pairs, Journal für die reine und angewandte Mathematik, Volume 547 (2002), pp. 51-68 | MR | Zbl

[BHM] F. Blanchard; B. Host; A. Maass Topological complexity, Ergod. Th. and Dynam. Sys, Volume 20 (2000), pp. 641-662 | MR | Zbl

[BL] F. Blanchard; Y. Lacroix Zero-entropy factors of topological flows, Proc. Amer. Math. Soc, Volume 119 (1993), pp. 85-992 | MR | Zbl

[DGS] M. Denker; C. Grillenberger; C. Sigmund Ergodic theory on compact spaces, Lecture Notes in Math, 527, Springer-Verlag, New York | Zbl

[F] H. Furstenberg Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Math. System Th., Volume 1 (1967), pp. 1-55 | MR | Zbl

[Fe] S. Ferenczi Measure-theoretic complexity of ergodic systems, Israel J. Math, Volume 100 (1997), pp. 189-207 | MR | Zbl

[G] T.N.T Goodman Topological sequence entropy, Proc. London Math. Soc, Volume 29 (1974), pp. 331-350 | MR | Zbl

[G1] E. Glasner A simple characterization of the set of μ-entropy pairs and applications, Israel J. Math, Volume 102 (1997), pp. 13-27 | MR | Zbl

[G2] E. Glasner Ergodic theory via joinings, Mathematical Surveys and Monographs, Volume 101 (2003) | MR | Zbl

[GW] E. Glasner; B. Weiss Strictly ergodic, uniform positive entropy models, Bull. Soc. Math. France, Volume 122 (1994) no. 3, pp. 399-412 | Numdam | MR | Zbl

[H-Y] W. Huang; S. Li; S. Shao; X. Ye Null systems and sequence entropy pairs, Ergod. Th. and Dynam. Sys, Volume 23-5 (2003), pp. 1505-1523 | MR | Zbl

[Hu] P. Hulse Sequence entropy and subsequence generators, J. London Math. Soc, Volume 26 (1982), pp. 441-450 | MR | Zbl

[HY] W. Huang; X. Ye Topological K-systems, a thrid approach (2001) (preprint)

[Kr] L. Kronecker Naherrungsweise ganzzahlige Auflosunglinear Gleichungen (Akad. Wiss. Werke III(1)), Volume 1179-93, 1271-99, pp. 47-109

[Ku] A. G. Kushnirenko On metric invariants of entropy type, Russian Math. Surveys, Volume 22 (1967) no. 5, pp. 53-61 | MR | Zbl

[P] W. Parry Topics in Ergodic Theory, Cambridge Tracks in Mathematics, Cambridge-New York, 1981 | MR | Zbl

[S] A. Saleski Sequence entropy and mixing, J. of Math. Anal. and Appli., Volume 60 (1977), pp. 58-66 | MR | Zbl

[W] B. Weiss Multiple recurrence and doubly minimal systems, AMS Contemporary Mathematics, Volume 215 (1998), pp. 189-196 | MR | Zbl

[Wa] P. Walters An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl

Cité par Sources :