Devil’s carpet of topological entropy and complexity of global dynamical behavior. (English) Zbl 1030.37023

Summary: For bimodal maps the concept of an equal topological entropy class (ETEC) is established by the dual star products. All the infinitely many ETEC plateaus and single points are harmonically organized in the kneading parameter plane, they construct a multifractal devil’s carpet, which possesses a perfect subregion similarity and a dual central symmetry. The entropy devil’s carpet reveals the complexity of global dynamical behavior in the whole parameter plane of bimodal systems.


37E05 Dynamical systems involving maps of the interval
37B40 Topological entropy
Full Text: DOI


[1] Feigenbaum, M.J.; Feigenbaum, M.J., The universal metric properties of nonlinear transformations, J. stat. phys., J. stat. phys., 21, 669-706, (1979) · Zbl 0515.58028
[2] Collet P, Eckmann JP. Iterated maps on the interval as dynamical systems. In: Jaffe A, Ruelle D, editors. Progress in physics, vol. 1. Boston: Birkhäuser; 1980 · Zbl 0458.58002
[3] Hao, B.-L., Elementary symbolic dynamics and chaos in dissipative systems, (1989), World Scientific Singapore · Zbl 0724.58001
[4] Hao B-L, Zheng W-M. Applied symbolic dynamics and chaos, Directions in chaos, vol. 7. Singapore: World Scientific; 1998
[5] Peng, S.-L.; Cao, K.-F., Global scaling behaviors and chaotic measure characterized by the convergent rates of period-p-tupling bifurcations, Phys. rev. E, 54, 3211-3220, (1996)
[6] Peng, S.-L.; Cao, K.-F.; Chen, Z.-X.; Erratum, Devil’s staircase of topological entropy and global metric regularity, Phys. lett. A, Phys. lett. A, 196, 378-443, (1995)
[7] Chen, Z.-X.; Cao, K.-F.; Peng, S.-L., Symbolic dynamics analysis of topological entropy and its multifractal structure, Phys. rev. E, 51, 1983-1988, (1995)
[8] Cao, K.-F.; Chen, Z.-X.; Peng, S.-L., Global metric regularity of the devil’s staircase of topological entropy, Phys. rev. E, 51, 1989-1995, (1995)
[9] Misiurewicz, M.; Szlenk, W., Entropy of piecewise monotone mappings, Studia math., 67, 45-63, (1980) · Zbl 0445.54007
[10] Newhouse, S.E., Continuity properties of entropy, Ann. math., Ann. math., 131, 409-410, (1990) · Zbl 0693.58009
[11] Milnor, J.; Tresser, C., On entropy and monotonicity for real cubic maps, Commun math. phys., 209, 123-178, (2000) · Zbl 0971.37007
[12] MacKay, R.S.; Tresser, C., Some flesh on the skeleton: the bifurcation structure of bimodal maps, Physica. D, 27, 412-422, (1987) · Zbl 0626.58038
[13] MacKay, R.S.; Tresser, C., Boundary of topological chaos for bimodal maps of the interval, J. London. math. soc. (2nd ser), 37, 164-181, (1988) · Zbl 0608.54016
[14] Mumbrú P. Estructura periòdica i entropia topològica de les aplicacions bimodals. PhD Thesis, Universitat Autònoma de Barcelona; 1987
[15] Llibre J, Mumbrú P. Extending the \(∗\)-product operator. In: Mira C, Netzer N, Simó C, Targonsky G, editors. Proceedings of the European Conference on Iteration Theory–1989. Singapore: World Scientific; 1991. p. 199-214 · Zbl 1026.37503
[16] Ringland, J.; Tresser, C., A genealogy for finite kneading sequences of bimodal maps on the interval, Trans. am. math. soc., 347, 4599-4624, (1995) · Zbl 0849.54033
[17] Chang, S.-J.; Wortis, M.; Wright, J.A., Iterative properties of a one-dimensional quartic map: critical lines and tricritical behavior, Phys. rev. A, 24, 2669-2684, (1981)
[18] MacKay, R.S.; Tresser, C., Transition to topological chaos for circle maps, Physica D, 19, 206-237, (1986) · Zbl 0596.58027
[19] Procaccia, I.; Thomae, S.; Tresser, C., First return maps as a unified renormalization scheme for dynamical systems, Phys. rev. A, 35, 1884-1900, (1987)
[20] Zheng, W.-M., Applied symbolic dynamics for the Lorenz-like map, Phys. rev. A, 42, 2076-2080, (1990)
[21] Peng, S.-L.; Zhang, X.-S., The generalized milnor – thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters, Commun math. phys., 213, 381-411, (2000) · Zbl 0974.37010
[22] Peng, S.-L.; Zhang, X.-S.; Cao, K.-F., Dual star products and metric universality in symbolic dynamics of three letters, Phys. lett. A, 246, 87-96, (1998) · Zbl 1044.37500
[23] Cao, K.-F.; Peng, S.-L., Complexity of routes to chaos and global regularity of fractal dimensions in bimodal maps, Phys. rev. E, 60, 2745-2760, (1999)
[24] Cvitanović, P., Invariant measurement of strange sets in terms of cycles, Phys. rev. lett., 61, 2729-2732, (1988)
[25] Artuso, R.; Aurell, E.; Cvitanović, P.; Artuso, R.; Aurell, E.; Cvitanović, P., Recycling of strange sets: II. applications, Nonlinearity, Nonlinearity, 3, 361-386, (1990) · Zbl 0702.58064
[26] Badii, R.; Politi, A., Complexity: hierarchical structures and scaling in physics. Cambridge nonlinear science series, (1997), Cambridge University Press Cambridge
[27] Milnor J, Thurston W. On iterated maps of the interval, I and II. Preprints (Princeton); 1977. In: Alexander JC, editor. Dynamical systems–Proceedings, University of Maryland 1986-1987. Lecture notes in mathematics, vol. 1342. Berlin: Springer-Verlag; 1988. p. 465-563
[28] Metropolis, N.; Stein, M.L.; Stein, P.R., On finite limit sets for transformations on the unit interval, J. comb. theory A, 15, 25-44, (1973) · Zbl 0259.26003
[29] Derrida, B.; Gervois, A.; Pomeau, Y., Iteration of endomorphisms on the real axis and representation of numbers, Ann. inst. Henri. Poincaré A, 29, 305-356, (1978) · Zbl 0416.28012
[30] Misiurewicz, M.; Nitecki, Z., Combinatorial patterns for maps of the interval, Memoirs am math soc, 456, R1etseq, (1991)
[31] Zheng, W.-M., Generalised composition law for symbolic itineraries, J. phys. A, 22, 3307-3313, (1989)
[32] Prog in Phys 1990;10:316-373(in Chinese)
[33] Falconer, K., Fractal geometry: mathematical foundations and applications, (1990), Wiley Chichester, Chapter 2 · Zbl 0689.28003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.