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 machine scientists, and graduate scholars attracted to fields re lated to likelihood thought, ergodic conception, symbolic and topological dynam ics. we wish to specific our gratitude to all of the members 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 medical recommend. VVe are looking to thank particularly 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 was once necessary to manage this assembly: ECOS-CONICYT, FONDAP application 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 education of the assembly in addition to for the significant job of unifying the typography of the several chapters of this book.
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And Weiss, B. 9B. Badoian, L. , Simple Connectivity of the Markov Partition Space, Pacific J. , to appear. Boyle, M. (1993) Symbolic Dynamics and Matrices, in Combinatorial and GraphTheoretical Problems in Linear Algebra, IMA Volumes in Math. 50, pp. 1-38. POSITIVE K-THEORY AND SYMBOLIC DYNAMICS 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 51 Boyle, M. (2000) Algebraic Aspects of Symbolic Dynamics, in Topics in Symbolic Dynamics and Applications (Temuco, 1997) London Math.
7. 1. DEFINITIONS Let S be a ring containing a semiring S+ which contains the additive and multiplicative identities 0 and 1. Let E(S+) be the set of N x N matrices E which agree with the identity matrix I in all except at most one entry E(i,j), with i -1= j and E(i,j) E S+. POSITIVE K-THEORY AND SYMBOLIC DYNAMICS 45 Let M be a collection of N x N matrices over S which differ from the identity matrix in at most finitely many entries. We will define a category Q:(M, S+) whose objects are M. In this category, we define a forward elementary positive equivalence to be a triple (C, (U, V), D) such that - C and D are in M; - UCV=D; - one of U and V is I; - U and V are in E(S+) A backward elementary positive equivalence is a triple (D, (U- I , V-I), C) such that (C, (U, V), D) is a forward elementary isomorphism.
The proof then follows from the fact that every cylinder [wo, . 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, .
Dynamics and Randomness by Valentin Afraimovich, Jesus Urias (auth.), Alejandro Maass, Servet Martínez, Jaime San Martín (eds.)