By Valentin Afraimovich, Jesus Urias (auth.), Alejandro Maass, Servet Martínez, Jaime San Martín (eds.)
This publication comprises the lectures given on the convention on Dynamics and Randomness held on the Centro de Modelamiento Matematico of the Universidad de Chile from December eleventh to fifteenth, 2000. This assembly introduced jointly mathematicians, theoretical physicists and theoretical computing device scientists, and graduate scholars attracted to fields re lated to chance idea, ergodic thought, symbolic and topological dynam ics. we want to precise our gratitude to the entire contributors of the con ference and to the folk who contributed to its association. particularly, to Pierre Collet, Bernard Host and Mike Keane for his or her clinical recommend. VVe are looking to thank specifically the authors of every bankruptcy for his or her good ready manuscripts and the stimulating meetings they gave at Santiago. we're additionally indebted to our sponsors and aiding associations, whose curiosity and aid used to be necessary to arrange this assembly: ECOS-CONICYT, FONDAP software in utilized arithmetic, French Cooperation, Fundacion Andes, Presidential Fellowship and Universidad de Chile. we're thankful to Ms. Gladys Cavallone for his or her very good paintings in the course of the guidance of the assembly in addition to for the enormous activity of unifying the typography of different chapters of this book.
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14. 15. 16. 17. 18. 19. 20. 2l. 22. 23. 24. 25. 26. 27. 28. VALENTIN AFRAIMOVICH AND JESUS URIAS Afraimovich, V. and Zaslavsky, G. M. 511, pp. 59-82. , and Schmeling, J. 149, pp. 755-783. Benkadda, S. M. 5ll. Bowen, R. (1979) Hausdorff Dimension of Quasi-Circles. Publ. Math. 50, pp. 259-273. Bohr, T. and Rand, D. 25D, pp. 387-398. Bruin, H. -M. Gambaudo, P. Hubert, P. Tisseur and S. ) World Scientific, pp. 117-124. -R. and Saussol, B. (2000) Sur les Dimensions Locales et les Dimensions des Mesures, Preprint.
3. The following theorem was proved in  by using the Shannon-McMillanBreiman theorem. L is ergodic with respect to a measurable transformation 0' : 0 --+ O. e. wE O. L be an ergodic measure of positive entropy on the weakly specified subshift (X, T), and ~ the finite partition of X defined in the setup. e. x E X. The following examples show that for systems with zero entropy this might be not true. 2. 5. Indeed, let ~o be a partition of Op by m-cylinders [wo, . ,Wm-l]. Since T- 1 ([wo, . , Wm-l]) = [w~, ...
Math. 50, pp. 259-273. Bohr, T. and Rand, D. 25D, pp. 387-398. Bruin, H. -M. Gambaudo, P. Hubert, P. Tisseur and S. ) World Scientific, pp. 117-124. -R. and Saussol, B. (2000) Sur les Dimensions Locales et les Dimensions des Mesures, Preprint. McCluskey, H. and Manning, A. (1983) Hausdorff Dimension for Horseshoes, Erg. Th. and Dynam. 3, pp. 251-260. de Melo, W. and van Strien, S. (1993) One-Dimensional Dynamics, Springer Verlag, Heidelber New York. Frostman, O. (1935) Potential d'Equilibre et Capacite des Ensembles avec quelques Applicications it la Theorie des Fonctions, Meddel.