Generalized entropies of chaotic maps and flows: A unified approach

Chaos

Volumn 7, Issue 4, 1997, Pages 694-700

  Badii, R ^a  

^a PAUL SCHERRER INSTITUTE   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0009017320     PISSN: 10541500     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.166267     Document Type: Article

References (34)

1. E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, England, 1993).
2. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai, Ergodic Theory (Springer, New York, 1982).
3. Ya. B. Pesin, Russ. Math. Surv. 32, 55 (1977).
4. D. Ruelle, Chaotic Evolution and Strange Attractors (Cambridge University Press, Cambridge, England 1989).
5. A. Renyi, Probability Theory (North-Holland, Amsterdam, 1970).
6. R. L. Adler, A. G. Konheim, and M. H. McAndrew, Trans. Am. Math. Soc. 114, 309 (1965).
7. Ya. G. Sinai, Russ. Math. Surv. 27, 21 (1972).
8. D. Ruelle, Thermodynamic Formalism, Vol. 5 of Encyclopedia of Mathematics and its Applications (Addison-Wesley, Reading, MA, 1978).
9. R. Bowen, Lect. Notes Math. 470, (1975).
10. R. Badii, Riv. Nuovo Cimento 12, 1 (1989).
11. T. Bohr and T. Tél, in Directions in Chaos, edited by B.-L. Hao (World Scientific, Singapore, 1988), Vol. 2, p. 194.
12. C. Beck and F. Schlögl, Thermodynamics of Chaotic Systems (Cambridge University Press, Cambridge, England, 1993).
13. P. Grassberger, R. Badii, and A. Politi, J. Stat. Phys. 51, 135 (1988).
14. A. Politi, R. Badii, and P. Grassberger, J. Phys. A 21, L736 (1988).
15. N. J. Balmforth, E. A. Spiegel, and C. Tresser, Phys. Rev. Lett. 72, 80 (1994).
16. L. Block, J. Keesling, S. Li, and K. Peterson, J. Stat. Phys. 55, 929 (1989).
17. M. I. Malkin, A. L. Zheleznyak, and I. L. Zheleznyak, Nonlinearity 4, 27 (1991).
18. E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963).
19. S. Newhouse and T. Pignataro, J. Stat. Phys. 72, 1331 (1993).
20. R. Badii, Phys. Rev. E 54, 4496 (1996).
21. P. Grassberger, T. Schreiber, and C. Schaffrath, Int. J. Bifurcation Chaos 1, 521 (1991).
22. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 2nd ed. (Springer, New York, 1986).
23. J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
24. V. M. Alekseev and M. V. Yakobson, Phys. Rep. 75, 287 (1981).
25. K. Petersen, Ergodic Theory (Cambridge University Press, Cambridge, England, 1989).
26. R. Badii, E. Brun, M. Finardi, L. Flepp, R. Holzner, J. Parisi, C. Reyl, and J. Simonet, Rev. Mod. Phys. 66, 1389 (1994).
27. M. Pollicott, in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, edited by T. Bedford et al. (Oxford University Press, Oxford, 1991) p. 153.
28. Similar extensions have been considered in Ref. 27 in the study of the topological entropy of the set of closed geodesics on compact manifolds with negative curvature. In that context, one obtains "Poincaré series" which are just the grand canonical sum (4.2) for q = 0.
29. R. Badii, in Chaotic Dynamics, Theory and Practice, edited by T. Bountis (Plenum. New York. 1992), p. 1.
30. R. Badii and A. Politi, Complexity: Hierarchical Structures and Scaling in Physics (Cambridge University Press, Cambridge, England, 1997).
31. M. J. Feigenbaum, M. H. Jensen, and I. Procaccia, Phys. Rev. Lett. 57, 1503 (1986).
32. M. J. Feigenbaum, J. Stat. Phys. 52, 527 (1988).
33. O. E. Rössler, Phys. Lett. A 57, 397 (1976).
34. In this way, intersections with decreasing (increasing) x are taken when x > 0 (x < 0) and the resulting Poincaré map has the same symmetry as the flow itself (see Ref. 26 for more details).