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Mixing properties of nearly maximal entropy measures for $ℤ^{d}$ shifts of finite type

100%

E. Arthur Robinson, Jr. , Şahin A. A.

nr 1

43-50

We prove that for a certain class of $ℤ^d$ shifts of finite type with positive topological entropy there is always an invariant measure, with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.

Słabe łamanie ergodyczności vs. determinizm

75%

Fuliński A.

nr 59

83-100

All physical processes are deterministic de iure. Physicists speak about different types of determinism of physical processes, depending on the degree with which their course can be anticipated. Usually, the course of ergodic processes can be predicted with less certainty than the non-ergodic ones, the latter being integrable. Recent measurements of motions of single particles in composite systems, especially in living biological cells, show that such motions are, in most cases, breaking the Boltzmann’s ergodic hypothesis. On the other hand, their trajectories are random, i.e., one cannot know a priori where the particle will be even in near future. This leads to conclusion that many existing in nature processes are nonergodic but not integrable, therefore predictable only in the mean, representing still other type of determinism.