Dependence

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. Invariant and remote events and the idea of ergodicity are introduced and the ergodic theorem is proved. The final sections introduce the contrasting idea of dependence on sub-σ‎-fields with the notions of regularity and strong and uniform mixing defined among other variants.

DENSITY OF MEASURE-THEORETIC DIRECTIONAL ENTROPY FOR LATTICE DYNAMICAL SYSTEMS

The density of the measure-theoretic directional entropy for the lattice dynamical system is introduced and it is shown that for a lattice dynamical transformation commuting with the shift transformation the density coincides with the entropy of the ℤ2-action generated by these two transformations, i.e. it is a constant function with respect to the direction. It is also proved that in the noncommutative case this result fails to be true.

C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems

A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. InDoc. Math.7 (2002) 1–30, the author constructed aC*-algebraO£associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebraO£, under a certain condition on £ called (II). As a result, the class of theC*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.