Periods and entropy for Lorenz-like maps. (English) Zbl 0678.34047

Alsedà, L.; Llibre, J.; Misiurewicz, M.; Tresser, Charles

Ann. Inst. Fourier 39, No. 4, 929-952 (1989).

We characterize the set of periods and its structure for the Lorenz-like maps depending on the rotation interval. Also, for these maps we give the best lower bound of the topological entropy as a function of the rotation interval.

Reviewer: L.Alseta

MSC:

37-XX	Dynamical systems and ergodic theory
54H20	Topological dynamics (MSC2010)

Keywords:

renormalization; Lorenz-like maps; the best lower bound; topological entropy; rotation interval

References:

[1]	[ALMM] , , and , Lower bounds for the topological entropy of continuous maps of the circle of degree one, Nonlinearity, 1 (1988), 463-479. · Zbl 0663.54023
[2]	[ALMS] , , and , Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point, Ergod. Th. and Dynam. Sys., 5 (1985), 501-518. · Zbl 0592.54037
[3]	[ALM] , and , Periodic orbits of maps of Y, Trans. Amer. Math. Soc., 313 (1989), 475-538. · Zbl 0803.54032
[4]	[BGMY] , , and , Periodic points and topological entropy of one-dimensional maps, Springer, Lect. Notes in Math., 819 (1980), 18-39. · Zbl 0447.58028
[5]	[CGT] , and , Une remarque sur la structure des endomorphismes de degré 1 du cercle, C.R. Acad. Sc. Paris, 299, Sér. I (1984), 145-148. · Zbl 0584.58004
[6]	[GPTT] , , and , New universal scenarios for the onset of chaos in Lorenz type flows, Phys. Rev. Lett., 57 (1986), 925-928.
[7]	[GT] and , Dynamique régulière ou chaotique. Applications du cercle ou de l’intervalle ayant une discontinuité, C.R. Acad. Sc., Paris, 300, Ser. I (1985), 311-313. · Zbl 0595.58031
[8]	[G1] , A strange, strange attractor, in the Hopf Bifurcation and its applications, Eds. J. E. Marsden and M. McCracken, Appl. Math. Sc. (Springer), 19 (1976), 368-381.
[9]	[G2] , Bifurcations of Dynamical Systems in : Dynamical Systems C.I.M.E. Lectures, Progress in Mathematics, 8, Birkhäuser, Boston, 1980. · Zbl 0451.58025
[10]	[H1] , The maximal measure for linear mod. one transformations, J. London Math. Soc., 23 (1981), 92-112. · Zbl 0431.54025
[11]	[H2] , Periodic points for piecewise monotonic transformation, Ergod. Th. and Dynam. Sys., 5 (1985), 237-256. · Zbl 0572.54036
[12]	[L] , Deterministic non-periodic flow, J. Atmos. Sci., 20 (1963), 130-141. · Zbl 1417.37129
[13]	[M1] , Periodic points of maps of degree one of a circle, Ergod Th. and Dynam. Sys., 2 (1982), 221-227. · Zbl 0508.58038
[14]	[M2] , Twist sets for maps of a circle, Ergod. Th. and Dynam. Sys., 4 (1984), 391-404. · Zbl 0573.58019
[15]	[M3] , Rotation intervals for a class of maps of the real line into itself, Ergod. Th. and Dynam. Sys., 6 (1986), 117-132. · Zbl 0615.54030
[16]	[MS] and , Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. · Zbl 0445.54007
[17]	[MT] and , On iterated maps of the interval, in Dynamical Systems, Ed. J. C. Alexander, Lecture Notes in Math., 1342 (1988), 465-563. · Zbl 0664.58015
[18]	[P] , The Lorenz attractor and a related population model, Lect. Notes in Math., Springer, 729 (1979), 169-187. · Zbl 0431.92022
[19]	[R] , Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., 8 (1957), 477-493. · Zbl 0079.08901
[20]	[RT] and , Rotation numbers for monotone functions on the circle, J. London Math. Soc., 34 (1986), 360-368. · Zbl 0623.58008