Commutativity and non-commutativity of topological sequence entropy
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, p. 1693-1709
Dans cet article nous étudions la propriété de commutativité pour l’entropie séquentielle topologique. Nous prouvons que si X est un espace métrique compact et f,g:XX sont deux fonctions continues, alors h A (fg)=h A (gf) pour toute suite croissance AX=[0,1] et nous construisons un contre-exemple dans le cas général. Au passage, nous prouvons aussi que l’égalité h A (f)=h A (f| n0 f n (X) ) est vraie si X=[0,1] mais ne l’est pas nécessairement si X est un espace métrique compact arbitraire.
In this paper we study the commutativity property for topological sequence entropy. We prove that if X is a compact metric space and f,g:XX are continuous maps then h A (fg)=h A (gf) for every increasing sequence A if X=[0,1], and construct a counterexample for the general case. In the interim, we also show that the equality h A (f)=h A (f| n0 f n (X) ) is true if X=[0,1] but does not necessarily hold if X is an arbitrary compact metric space.
@article{AIF_1999__49_5_1693_0,
     author = {Balibrea, Francisco and Pe\~na, Jose Salvador C\'anovas and L\'opez, V\'\i ctor Jim\'enez},
     title = {Commutativity and non-commutativity of topological sequence entropy},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     pages = {1693-1709},
     doi = {10.5802/aif.1735},
     zbl = {0990.37010},
     mrnumber = {2001g:37015},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_5_1693_0}
}
Balibrea, Francisco; Peña, Jose Salvador Cánovas; López, Víctor Jiménez. Commutativity and non-commutativity of topological sequence entropy. Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1693-1709. doi : 10.5802/aif.1735. https://www.numdam.org/item/AIF_1999__49_5_1693_0/

[1] R. L. Adler, A. G. Konheim and M. H. Mcandrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. | MR 30 #5291 | Zbl 0127.13102

[2] F. Balibrea, J. S. Cánovas Peña and V. Jiménez López, Some results on entropy and sequence entropy, Internat. J. Bifur. Chaos Appl. Sci. Engrg. (to appear). | Zbl 0942.28017

[3] F. Balibrea, J. S. Cánovas Peña and V. Jiménez López, Topological sequence entropy on the nonwandering set can be less than on the whole space: an interval counterexample, preprint.

[4] R. A. Dana and L. Montrucchio, Dynamic complexity in duopoly games, J. Econom. Theory, 44 (1986), 40-56. | MR 87k:90045 | Zbl 0617.90104

[5] N. Franzová and J. Smital, Positive sequence topological entropy characterizes chaotic maps, Proc. Amer. Math. Soc., 112 (1991), 1083-1086. | MR 91j:58107 | Zbl 0735.26005

[6] T. N. T. Goodman, Topological sequence entropy, Proc. London Math. Soc., 29 (1974), 331-350. | MR 50 #8482 | Zbl 0293.54043

[7] W. H. Gottschalk and G. A. Hedlung, Topological Dynamics, Amer. Math. Soc., 1955. | Zbl 0067.15204

[8] V. Jiménez López, An explicit description of all scrambled sets of weakly unimodal functions of type 2∞, Real. Anal. Exch., 21 (1995/1996), 1-26. | MR 97g:58109 | Zbl 0879.58044

[9] S. Kolyada and L'. Snoha, Topological entropy of nonautononous dynamical systems, Random and Comp. Dynamics, 4 (1996), 205-233. | MR 98f:58126 | Zbl 0909.54012

[10] A. Linero, Cuestiones sobre dinámica topológica de algunos sistemas bidimensionales y medidas invariantes de sistemas unidimensionales asociados, PhD thesis, Universidad de Murcia, 1998.

[11] M. Misiurewicz and J. Smital, Smooth chaotic functions with zero topological entropy, Ergod. Th. and Dynam. Sys., 8 (1988), 421-424. | MR 90a:58118 | Zbl 0689.58028

[12] W. Szlenk, On weakly* conditionally compact dynamical systems, Studia Math., 66 (1979), 25-32. | MR 81a:54043 | Zbl 0497.54039

[13] P. Walters, An introduction to ergodic theory, Springer-Verlag, Berlin, 1982. | MR 84e:28017 | Zbl 0475.28009