Geodesic
Continuity of topological entropy for piecewise smooth Lorenz type mappings
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 59-66 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For one-dimensional mappings of Lorenz type, the problem on behavior of the topological entropy as the function of a mapping is studied. In the previous paper the authors proved that entropy as the function of a mapping with $C^0$-topology can have jumps only for exceptional case, namely, in a neighbourhood of a mapping with zero entropy, and moreover, if and only if two kneading invariants are periodic with the same period. In the present paper we show that for the class of Lorenz mappings having zero one-sided derivatives at the discontinuity point and with $C^1$-topology, such an exceptional case is impossible, and thus the entropy depends continously on the mapping.
Keywords: topological entropy, Lorenz type mappings, kneading invariant. 
@article{SVMO_2016_18_2_a7,
     author = {M. I. Malkin and K. A. Saphonov},
     title = {Continuity of topological entropy for piecewise smooth {Lorenz} type mappings},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {59--66},
     year = {2016},
     volume = {18},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a7/}
}
TY  - JOUR
AU  - M. I. Malkin
AU  - K. A. Saphonov
TI  - Continuity of topological entropy for piecewise smooth Lorenz type mappings
JO  - Žurnal Srednevolžskogo matematičeskogo obŝestva
PY  - 2016
SP  - 59
EP  - 66
VL  - 18
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a7/
LA  - ru
ID  - SVMO_2016_18_2_a7
ER  - 
%0 Journal Article
%A M. I. Malkin
%A K. A. Saphonov
%T Continuity of topological entropy for piecewise smooth Lorenz type mappings
%J Žurnal Srednevolžskogo matematičeskogo obŝestva
%D 2016
%P 59-66
%V 18
%N 2
%U http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a7/
%G ru
%F SVMO_2016_18_2_a7
M. I. Malkin; K. A. Saphonov. Continuity of topological entropy for piecewise smooth Lorenz type mappings. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 2, pp. 59-66. http://geodesic.mathdoc.fr/item/SVMO_2016_18_2_a7/

[1] M. Malkin, “On continuity of entropy of discontinuous mappings of the interval”, Selecta Mathematica Sovietica, 8 (1989), 131–139 | Zbl

[2] M.I. Malkin, K.A. Safonov, “Tochnaya otsenka razryvov entropii dlya otobrazhenii lorentsevskogo tipa”, Zhurnal Srednevolzhskogo Matematicheskogo Obschestva, 17:4 (2015), 31–36 | Zbl

[3] L.-S. Young, “On the prevalence of horseshoes”, Trans. Amer. Math. Soc., 263:1 (1981), 75–88 | DOI | MR | Zbl

[4] Afraimovich V., Sze-Bi Hsu, Lecture on chaotic dynamical systems, Studies in Advanced Mathematics, 28, AMS/IP, N.-Y., 2002 | MR

[5] M.-C. Li, M. Malkin, “Smooth symmetric and Lorenz models for unimodal maps”, International Journal of Bifurcation and Chaos, 13 (2003), 3353–3372 | DOI | MR

[6] J.Milnor, W.Thurston, “On iterated maps of the interval”, Dynamical Systems, Lec. Notes Math., 1342, ed. J.C. Alexander, Springer-Verlag, N.-Y., 1988 | MR

[7] M. Misiurewicz, “Jumps of entropy in one dimension”, Fundamenta Mathematicae, 132 (1989), 215–226 | DOI | MR | Zbl