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Inherent randomicity in 4-symbolic dynamics. (English) Zbl 1083.37502

Summary: The inherent randomicity in 4-symbolic dynamics is clarified in this paper. The symbolic sequences bear three characteristics. The distribution of frequency, inter-occurrence times and the alignment of two random sequences are amplified in detail. By using transfer probability of Markov chains, we obtain analytic expressions of generating functions in four probabilities stochastic wander models, which can be applied to all 4-symbolic systems. We hope to offer a symbolic platform that satisfies these stochastic properties and to study some properties of DNA sequences.

MSC:

37B10 Symbolic dynamics
92D10 Genetics and epigenetics
37N25 Dynamical systems in biology
60C05 Combinatorial probability
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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[1] Ulam, S.M.; Neumann, J.V., On combination of stochastic and deterministic processes, Bull am math soc, 53, 1120, (1947)
[2] Hao, B.L., Elementary symbolic dynamics and chaos in dissipative systems, (1989), World Scientific Singapore · Zbl 0724.58001
[3] Hao, B.L., Symbolic dynamics and characterization of complexity, Physica D, 51, 161-176, (1991) · Zbl 0744.58014
[4] Hao, B.L.; Zheng, W.M., Symbolic dynamics and chaos, Directions in chaos, vol. 7, (1998), World Scientific Singapore
[5] Collet, P.; Eckmann, J.P., Iterated maps on the interval as dynamical systems, (1980), Birkhäuser Boston · Zbl 0458.58002
[6] Alekseev, V.M.; Yakobson, M.V., Symbolic dynamics and hyperbolic dynamics systems, Phys rep, 75, 287-325, (1981)
[7] Bowen, R., ()
[8] Xie, H.M., On formal languages of one-dimensional dynamics systems, Nonlinearity, 6, 997-1007, (1993) · Zbl 0803.68064
[9] Xie, H.M., Grammatical complexity and one-dimensional dynamics systems, Directions in chaos, vol. 6, (1996), World Scientific Singapore
[10] Peng, S.L.; Luo, L.S., The ordering of critical periodic points in coordinate space by symbolic dynamics, Phys lett A, 153, 345-352, (1991)
[11] Zhou, Z.; Peng, S.L., Cyclic star products and universalities in symbolic dynamics of trimodal maps, Physica D, 140, 213-226, (2000) · Zbl 0982.37027
[12] Coelho, Z.; Collet, P., Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle, J prob theory relat fields, 99, 237-250, (1994) · Zbl 0808.60028
[13] Coelho, Z., On discrete stochastic processes generated by deterministic sequences and multiplication machines, Indag math NS, 11, 3, 359-378, (2000) · Zbl 1003.37002
[14] 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, 4, 3211-3220, (1996)
[15] Billingsley, P., Probability and measure, (1986), John Wiley & Sons New York · Zbl 0649.60001
[16] Shi, J.X.; Cao, K.F.; Guo, T.L.; Peng, S.L., Metric universality for the devil’s staircase of topological entropy, Phys lett A, 211, 1, 25-28, (1996) · Zbl 1073.37515
[17] Zhang, X.S.; Liu, X.D.; Kwek, K.H.; Peng, S.L., Disorder versus order: global multifractal relationship between topological entropies and universal convergence rates, Phys lett A, 211, 3, 148-154, (1996) · Zbl 1073.37509
[18] Cao, K.F.; Chen, Z.X.; Peng, S.L., Global metric regularity of the devil’s staircase of topological entropy, Phys rev E, 51, 3, 1989-1995, (1995)
[19] Chen, Z.X.; Cao, K.F.; Peng, S.L., Symbolic dynamics analysis of topological entropy and its multifractal structure, Phys rev E, 51, 3, 1983-1988, (1995)
[20] Peng, S.L.; Cao, K.F.; Chen, Z.X., Devil’s staircase of topological entropy and global metric regularity, Phys lett A, 193, 5-6, 437-443, (1994) · Zbl 0961.37504
[21] Chen, Z.X.; Zhou, Z., Entropy invariants: I. the universal order relation of order-preserving star products, Chaos, solitons & fractals, 15, 4, 713-727, (2003) · Zbl 1041.37014
[22] Chen, Z.X.; Zhou, Z., Entropy invariants: II. the block structure of Stefan matrices, Chaos, solitons & fractals, 15, 4, 729-742, (2003) · Zbl 1041.37015
[23] Liang, X.; Jiang, J.F., On the topological entropy, nonwandering set and chaos of monotone and competitive dynamical systems, Chaos, solitons & fractals, 14, 689-696, (2002) · Zbl 0998.37004
[24] Hao, B.L., Fractals from genomes-exact solutions of a biology-inspired problem, Physica A, 282, 225-246, (2000)
[25] Hao, B.L.; Lee, H.C.; Zhang, S.Y., Fractals related to long DNA sequences and complete genomes, Chaos, solitons & fractals, 11, 825-836, (2000) · Zbl 0959.92019
[26] Bershadskii, A., Multifractal and probabilistic properties of DNA sequences, Phys lett A, 284, 136-140, (2001) · Zbl 0991.92012
[27] Grimm, H.; Rupprecht, A., Low frequency dynamics of DNA, Physica B, 234-236, 183-187, (1997)
[28] Allegrini, P.; Grigolini, P.; West, B.J., A dynamical approach to DNA sequences, Phys lett A, 211, 217-222, (1996) · Zbl 1060.92506
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