Counterexamples To The Product Rule For Entropy

摘要

The formularnh(T_1 × T_2) = h(T_1) + h(T_2)rnfor the entropy of the direct product of T_1 and T_2 holds both for measure-theoretic entropy, when T_1 and T_2 are measure-preserving transformations on probability spaces, and for topological entropy, when T_1 and T_2 are continuous transformations on compact topological spaces. Examples are given to show that the formula does not extend to sequence entropy in either case. Moreover, in the topological case, the formula does not hold, in general, when the spaces are non-compact.