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On the countable generator theorem

Michael Keane, Jacek Serafin (1998)

Fundamenta Mathematicae

Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T.

On the directional entropy of ℤ²-actions generated by cellular automata

M. Courbage, B. Kamiński (2002)

Studia Mathematica

We show that for any cellular automaton (CA) ℤ²-action Φ on the space of all doubly infinite sequences with values in a finite set A, determined by an automaton rule F = F [ l , r ] , l,r ∈ ℤ, l ≤ r, and any Φ-invariant Borel probability measure, the directional entropy h v ( Φ ) , v⃗= (x,y) ∈ ℝ², is bounded above by m a x ( | z l | , | z r | ) l o g A if z l z r 0 and by | z r - z l | in the opposite case, where z l = x + l y , z r = x + r y . We also show that in the class of permutative CA-actions the bounds are attained if the measure considered is uniform Bernoulli.

On the entropy and generators of dynamical systems

Beloslav Riečan (1996)

Applications of Mathematics

Recently D. Dumitrescu ([4], [5]) introduced a new kind of entropy of dynamical systems using fuzzy partitions ([1], [6]) instead of usual partitions (see also [7], [11], [12]). In this article a representation theorem is proved expressing the entropy of the dynamical system by the entropy of a generating partition.

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