Question 1: Let S be a transitive constant-to-one extension of the subshift of finite. De-Finettis theorem is instrumental in statistical modeling. Constant-to-one extensions have also been given some attention in symbolic. : SN such that mDom () mDom () for all K is an average of Bernoulli measures. The corresponding statement for sofic systems was proved to be false, also in. weakly mixing, constant-to-one extension of a Bernoulli system is also Bernoulli. In, one of the authors has shown that transitive constant-to-one extensions of coded systems (defined in ) are coded, or equivalently, according to Krieger's characterization, are the limit of an increasing sequence of transitive subshifts of finite type. Keywords : pascal triangle complexity entropy weak mixing bernoulli. We adopt this point of view and define subshifts of finite type via the Parry-Bowen-Lanford matrix, A. This matrix essentially dates from the earlier papers of Parry 9, 10 on intrinsic Markov chains. Constant-to-one extensions have also been given some attention in Symbolic Dynamics: in, Nasu investigated constant-to-one factor maps between subshifts of finite type and proved that they are necessarily right- and leftclosing. We construct a representation of thesystem by a subshift on a two-symbol alphabet. A (see 1) to each subshift of finite type. Chen, Some Nonrobust Bernolli-Shift Rules, International Journal of Bifurcation and Chaos, Vol. 1 Introduction In the setting of metric ergodic theory, Rudolph proved that any weakly mixing constant-to-one extension of a Bernoulli system is also Bernoulli. The necessity of the former condition was proved by Kolmogorov 21, 22 while Ornstein 32 1. Provided a constant-to-one extension of a subshift of finite type is sofic and transitive, then it is also of finite type. The notion of Ornsteins d-distance made possible to demonstrate the Kolmogorov-Ornstein Isomorphism Theorem which establish the following statement: two Bernoulli shifts are isomorphic if and only if they have the same entropy. Yang, Chaos Caused by a Topologically Mixing Map, in Dynamical Systems and Related Topics ( World Scientific, Singapore, 1992 ). Wolfram, A New Kind of Science ( Wolfram Media, Champaign, Illinois, USA, 2002 ). Wolfram, Theory and Application of Cellular Automata ( World Scientific, Singapore, 1986 ). Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos ( Springer-Verlag, Berlin, NY, 1990 ). Ohi, Chaotic properties of the elementary cellular automata rule 168 in Wolfram's class I, AUTOMATA-2008: Theory and Applications of Cellular Automata ( Luniver Press, UK, 2008 ). Baba, Introduction to Chaos ( Institute of Physics Publishing, Bristol, 1999 ). We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g-measure. In addition, we prove that these unique g-measures have Bernoulli natural extensions. Kitchens, Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts ( Springer-Verlag, Berlin, NY, 1998 ). We improve and subsume the conditions of Johansson and Oberg and Berbee for uniqueness of a g-measure, i.e., a stationary distribution for chains with complete connections. für die reine und Angewandte Mathematik 547, 51 (2002). Small systems in a thermodynamic medium - like colloids in a suspension or the molecular machinery in living cells - are strongly affected by the thermal fluctuations of their environment.

Bernoulli subshift