Entropy computing via integration over fractal measures

Chaos

Volumn 10, Issue 1, 2000, Pages 180-188

  Słomczyński, Wojciech ^a   Kwapień, Jarosław ^b   Zyczkowski, Karol ^a,c  

^a JAGIELLONIAN UNIVERSITY   (Poland)

^b INSTITUTE OF NUCLEAR PHYSICS   (Poland)

^c CENTER FOR THEORETICAL PHYSICS   (Poland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034147673     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166492     Document Type: Article

References (63)

1. J.-P. Eckmann and D. Ruelle, "Ergodic theory of chaos and strange attractors," Rev. Mod. Phys. 57, 617-653 (1985).
2. A. Cohen and I. Procaccia, "Computing the Kolmogorov entropy from time signals of dissipative and conservative dynamical systems," Phys. Rev. A 31, 1872-1882 (1985).
3. P. Schuster, Deterministic Chaos. An Introduction (VCH Verlagsgesellschaft, Weinheim, 1988).
4. W. Słomczyński and K. Zyczkowski, "Quantum chaos, an entropy approach," J. Math. Phys. 35, 5674-5700 (1994); 36, 5201 (1995).
5. W. Słomczyński and K. Zyczkowski, "Quantum chaos, an entropy approach," J. Math. Phys. 35, 5674-5700 (1994); 36, 5201 (1995).
6. K. Zyczkowski and W. Słomczyński, "Coherent states quantum entropy," in Proceedings of the International Conference on Dynamical Systems and Chaos, Tokyo, 23-27 May, 1994, Vol 2, edited by Y. Aizawa et al. (World Scientific, Singapore, 1995), pp. 467-470.
7. J. Kwapień, W. Słomczýski, and K. Zyczkowski, "Coherent states measurement entropy," J. Phys. A 30, 3175-3200 (1997).
8. W. Słomczýski and K. Zyczkowski, "Mean dynamical entropy of quantum maps on the sphere diverges in the semiclassical limit." Phys. Rev. Lett. 80, 1880-1883 (1998).
9. A. Lasota and J. Myjak, "Generic properties of fractal measures," Bull. Pol. Acad., Math. 42, 283-296 (1994).
10. T. Szarek, "Generic properties of learning systems," Ann. Polon. Math. (to be published).
11. M. F. Barnsley, Fractals Everywhere (Academic, Boston, 1988; 2nd ed., 1993).
12. I. Guarneri and G. Mantica, "Multifractal energy spectra and their dynamical implications," Phys. Rev. Lett. 73, 3379-3382 (1994).
13. J. Bene and P. Szépfalusy, "Multifractal properties in the one-dimensional random-field Ising model," Phys. Rev. A 37, 1703-1707 (1988).
14. G. Radons, "Emergence of quenched phases and second order transitions for sums of multifractal measures," Phys. Rev. Lett. 75, 2518-2521 (1995).
15. M. F. Barnsley, S. G. Demko, J. H. Elton, and J. S. Geronimo, "Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities," Ann. Inst. Henri Poincaré, Sect. A 24, 367-394 (1988); 25, 589 (1989).
16. M. F. Barnsley, S. G. Demko, J. H. Elton, and J. S. Geronimo, "Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities," Ann. Inst. Henri Poincaré, Sect. A 24, 367-394 (1988); 25, 589 (1989).
17. M. Iosifescu and G. Grigorescu, Dependence with Complete Connections and Its Applications (Cambridge University Press, Cambridge, 1990).
18. A. Csordás, G. Györgyi, P. Szépfalusy, and T. Tél. "Statistical properties of chaos demonstrated in a class of one-dimensional maps," Chaos 3, 31-50 (1993).
19. A. O. Lopes and Wm. D. Withers, "Weight-balanced measures and free energy for one-dimensional dynamics," Forum Math. 5, 161-182 (1993).
20. P. Góra and A. Boyarsky, "Iterated function systems and dynamical systems," Chaos 5, 634-639 (1995).
21. W. Słomczyński, "From quantum entropy to iterated function systems," Chaos Solitons Fractals 8, 1861-1864 (1997).
22. E. S. Abers and B. W. Lee, "Gauge theories," Phys. Rep. 9, 1-141 (1973).
23. T. Kaijser, "On a new contraction condition for random systems with complete connections," Rev. Roum. Math. Pures et Appl. 26, 1075-1117 (1981).
24. J. H. Elton, "An ergodic theorem for iterated maps," Ergod. Th. & Dynam. Sys. 7, 481-488 (1987).
25. M. Barnsley and J. E. Elton, "A new class of Markov processes for image encoding," Adv. Appl. Probab. 20, 14-32 (1988).
26. L. Arnold and H. Crauel, "Iterated function systems and multiplicative ergodic theory," in Diffusion Processes and Related Problems in Analysis, Vol. II, edited by M. Pinsky and V. Wihstutz (Birkhauser, Boston, 1992), pp. 283-305.
27. S. Grigorescu, "Limit theorems for Markov chains arising from iterated function systems," Rev. Roum. Math. Pures et Appl. 37, 887-899 (1992).
28. E. Gadde, Stable IFSs with probabilities. An ergodic theorem, Research Rep. 10, (1994) Dept. of Mathematics, Umeå University.
29. A. Lasota and J. A. Yorke, "Lower bound technique for Markov operators and iterated function systems," Random Comput. Dyn. 2, 41-77 (1994).
30. W. Jarczyk and A. Lasota, "Invariant measures for fractals and dynamical systems," Bull. Pol. Acad., Math. 43, 347-361 (1995).
31. A. Lasota, "From fractals to stochastic differential equations," in Chaos - The Interplay Between Stochastic and Deterministic Behaviour. edited by P. Garbaczewski et al. (Springer, Berlin, 1995), pp. 235-255.
32. K. Łoskot and R. Rudnicki, "Limit theorems tor stochastically perturbed dynamical systems," J. Appl. Probab. 32, 459-469 (1995).
33. B. Forte and F. Mendivil. "A classical ergodic property for IFS: a simple proof," Ergod. Th. & Dynam. Sys. 18, 609-611 (1998).
34. D. Silvestrov and Ö. Stenflo, "Ergodic theorems tor iterated function systems controlled by regenerative sequences," J. Theoret. Probab. 11, 589-608 (1998).
35. A. Öberg, Approximation of invariant measures for iterated function systems. U.U.D.M. Doctoral thesis 5 (1998), Dept. of Mathematics. Umeå University.
36. A. Edalat, "Power domains and iterated function systems," Inform. and Comput. 124, 182-197 (1996).
37. A. Lasota and M. Mackey, Chaos, Fractals and Noise (Springer, Berlin, 1994).
38. K. Zyczkowski and W. Słomczyński, "Exponential decay of relative entropies to the Kolmogorov-Sinai entropy for the standard map," Phys. Rev. E 52, 6879-6880 (1995).
39. Yu. Kifer, Ergodic Theory of Random Transformations (Birkhauser, Boston. 1986).
40. J. N. Kapur, Measures of Information and Their Applications (Wiley, New-York, 1994).
41. A. Rényi, "On measures of entropy and information," in Proceedings of the Fourth Berkeley Symp. Math. Statist. Prob. 1960. Vol. I (University of California Press. Berkeley, 1961). pp. 547-561.
42. F. Takens and E. Verbitski, "Generalized entropies: Rényi and correlation integral approach." Nonlinearity 11, 771-782 (1998).
43. D. Bessis, G. Paladin, G. Turchetti, and S. Vaienti, "Generalized dimensions, entropies, and Liapunov exponents from the pressure function for strange sets." J. Stat. Phys. 51, 109-134 (1988).
44. J.-M. Ghez, E. Orlandini, M.-C. Tesi, and S. Vaienti, "Dynamical integral transform on fractal sets and the computation of entropy," Physica D 63, 282-298 (1993).
45. W. Słomczyński, "Dynamical entropy and iterated function systems" (in preparation).
46. H. Hentschel and I. Procaccia. "The infinite number of generalized dimensions of fractals and strange attractors," Physica D 8, 435-444 (1983).
47. P. Grassberger and I. Procaccia. "On the characterization of strange attractors." Phys. Rev. Lett. 50, 346-349 (1983).
48. L. Olsen, Random Geometrically Graph Directed Self-Similar Multifractals (Longman, Harlow, 1994).
49. L. Olsen, "A multifractal formalism," Adv. Math. 116, 82-196 (1995).
50. K. Falconer, Techniques in Fractal Geometry (Wiley, Chichester, 1997).
51. W. Chin, B. Hunt, and J. A. Yorke, "Correlation dimension for iterated function systems." Trans. Am. Math. Soc. 349, 1783-1796 (1997).
52. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Schraiman, "Fractal measures and their singularities. The characterization of strange sets." Phys. Rev. A 33, 1141-1151 (1986).
53. T. Bohr and D. Rand, "The entropy function for characteristic exponents," Physica D 25, 387-398 (1987).
54. D. Rand, "The singularity spectrum f(α) for cookie-cutters," Ergod. Th. & Dynam. Sys. 9, 527-541 (1989).
55. L. Barreira, Y. Pesin, and J. Schmeling, "On a general concept of multifractality, multifractal spectra tor dimensions, entropies, and Lyapunov exponents. Multifractal rigidity," Chaos 7, 27-38 (1997).
56. I. Białynicki - Birula and D. Wójcik, private communication.
57. M. Holschneider, "Large-scale renormalization of Fourier-transforms of self-similar measures and self-similarity of Riesz measures." J. Math. Anal. Appl. 200, 307-314 (1996).
58. M. Fannes, B. Nachtergaele, and L. Słegers, "Functions of Markov processes and algebraic measures." Rev. Math. Phys. 4, 39-64 (1992).
59. P. Grassberger and I. Procaccia, "Estimation of the Kolmogorov entropy from a chaotic signal," Phys. Rev. A 28, 2591-2593 (1983).
60. K. Pawelzik and H. G. Schuster. "Generalized dimensions and entropies from a measured time series," Phys. Rev. A 35, 481-484 (1987).
61. K. Zyczkowski and E. Bolt, "On the entropy devil's staircase for the family of gap tent maps." Physica D 132, 392-410 (1999).
62. Y.-C. Lai, K. Zyczkowski, and C. Grebogi, "Universal behavior in the parametric evolution of chaotic saddles," Phys. Rev. E 59, 5261-5265 (1999).
63. A. Ostruszka, P. Pakoński, W. Słomczyński, and K. Zyczkowski, "Dynamical entropy for systems with stochastic perturbations," preprint chaodyn 9905041.