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 -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.
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 -tuple d’entropie séquentielle pour une mesure et nous prouvons que l’ensemble - tuple d’entropie sequentielle pour une mesure est contenu dans l’ensemble de -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.
Keywords: sequential entropy, complexity
Mot clés : entropie séquentielle, complexité
Huang, Wen 1; Maass, Alejandro ; Ye, Xiangdong 
@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, Volume 54 (2004) no. 4, pp. 1005-1028. doi : 10.5802/aif.2041. https://aif.centre-mersenne.org/articles/10.5802/aif.2041/
[B-R] Entropy pairs for a measure, Ergod. Th. and Dynam. Sys, Volume 15 (1995), pp. 621-632 | MR | Zbl
[B1] Fully positive topological entropy and topological mixing, Symbolic dynamics and its applications, AMS Contemporary Mathematics, Volume 135 (1992), pp. 95-105 | MR | Zbl
[B2] A disjointness theorem involving topological entropy, Bull. de la Soc. Math. de France, Volume 121 (1993), pp. 465-478 | Numdam | MR | Zbl
[Be] Ergodic Ramsey theory -- an update, Ergodic theory of actions (Warwick, 1993-1994) (London Math. Soc. Lecture Notes Ser.), Volume 228 (1996), pp. 1-61 | Zbl
[BGH] Variations on the variational principle, Ergod. Th. and Dynam. Sys, Volume 17 (1997), pp. 29-53 | MR | Zbl
[BGKM] On Li-Yorke pairs, Journal für die reine und angewandte Mathematik, Volume 547 (2002), pp. 51-68 | MR | Zbl
[BHM] Topological complexity, Ergod. Th. and Dynam. Sys, Volume 20 (2000), pp. 641-662 | MR | Zbl
[BL] Zero-entropy factors of topological flows, Proc. Amer. Math. Soc, Volume 119 (1993), pp. 85-992 | MR | Zbl
[DGS] Ergodic theory on compact spaces, Lecture Notes in Math, 527, Springer-Verlag, New York | Zbl
[F] Disjointness in ergodic theory, minimal sets and a problem in diophantine approximation, Math. System Th., Volume 1 (1967), pp. 1-55 | MR | Zbl
[Fe] Measure-theoretic complexity of ergodic systems, Israel J. Math, Volume 100 (1997), pp. 189-207 | MR | Zbl
[G] Topological sequence entropy, Proc. London Math. Soc, Volume 29 (1974), pp. 331-350 | MR | Zbl
[G1] A simple characterization of the set of -entropy pairs and applications, Israel J. Math, Volume 102 (1997), pp. 13-27 | MR | Zbl
[G2] Ergodic theory via joinings, Mathematical Surveys and Monographs, Volume 101 (2003) | MR | Zbl
[GW] Strictly ergodic, uniform positive entropy models, Bull. Soc. Math. France, Volume 122 (1994) no. 3, pp. 399-412 | Numdam | MR | Zbl
[H-Y] Null systems and sequence entropy pairs, Ergod. Th. and Dynam. Sys, Volume 23-5 (2003), pp. 1505-1523 | MR | Zbl
[Hu] Sequence entropy and subsequence generators, J. London Math. Soc, Volume 26 (1982), pp. 441-450 | MR | Zbl
[HY] Topological K-systems, a thrid approach (2001) (preprint)
[Kr] Naherrungsweise ganzzahlige Auflosunglinear Gleichungen (Akad. Wiss. Werke III(1)), Volume 1179-93, 1271-99, pp. 47-109
[Ku] On metric invariants of entropy type, Russian Math. Surveys, Volume 22 (1967) no. 5, pp. 53-61 | MR | Zbl
[P] Topics in Ergodic Theory, Cambridge Tracks in Mathematics, Cambridge-New York, 1981 | MR | Zbl
[S] Sequence entropy and mixing, J. of Math. Anal. and Appli., Volume 60 (1977), pp. 58-66 | MR | Zbl
[W] Multiple recurrence and doubly minimal systems, AMS Contemporary Mathematics, Volume 215 (1998), pp. 189-196 | MR | Zbl
[Wa] An introduction to ergodic theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl
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