Effective symbolic dynamics, random points, statistical behavior, complexity and entropy

@misc{citeulike:2796811, abstract = {We consider the dynamical behavior of Martin-L\"of random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sina\"i entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy.}, author = {Galatolo, Stefano and Hoyrup, Mathieu and Rojas, Cristobal }, citeulike-article-id = {2796811}, eprint = {0801.0209}, keywords = {computational-complexity, entropy, symbolic-dynamics}, month = {Apr}, posted-at = {2008-05-14 07:32:57}, priority = {2}, title = {Effective symbolic dynamics, random points, statistical behavior, complexity and entropy}, url = {http://arxiv.org/abs/0801.0209}, year = {2008} }

TY - ELEC ID - citeulike:2796811 L3 - citeulike-article-id:2796811 TI - Effective symbolic dynamics, random points, statistical behavior, complexity and entropy N2 - We consider the dynamical behavior of Martin-L\"of random points in dynamical systems over metric spaces with a computable dynamics and a computable invariant measure. We use computable partitions to define a sort of effective symbolic model for the dynamics. Through this construction we prove that such points have typical statistical behavior (the behavior which is typical in the Birkhoff ergodic theorem) and are recurrent. We introduce and compare some notions of complexity for orbits in dynamical systems and prove: (i) that the complexity of the orbits of random points equals the Kolmogorov-Sina\"i entropy of the system, (ii) that the supremum of the complexity of orbits equals the topological entropy. KW - computational-complexity KW - entropy KW - symbolic-dynamics AU - Galatolo, Stefano AU - Hoyrup, Mathieu AU - Rojas, Cristobal PY - 2008/Apr/29/ UR - http://arxiv.org/abs/0801.0209 ER -