On directional entropy functions Article

Received: 24 September 1997 Revised: 10 July 1998 DOI :
10.1007/BF02780179

Cite this article as: Park, K.K. Isr. J. Math. (1999) 113: 243. doi:10.1007/BF02780179
Abstract Given aZ ^{2} -process, the measure theoretic directional entropy function,h (\(\vec v\) % MathType!End!2!1!), is defined on\(S^1 = \left\{ {\vec v:\left\| {\vec v} \right\| = 1} \right\} \subset R^2 \) % MathType!End!2!1!. We relate the directional entropy of aZ ^{2} -process to itsR ^{2} suspension. We find a sufficient condition for the continuity of directional entropy function. In particular, this shows that the directional entropy is continuous for aZ ^{2} -action generated by a cellular automaton; this finally answers a question of Milnor [Mil]. We show that the unit vectors whose directional entropy is zero form aG _{δ} subset ofS ^{1} . We study examples to investigate some properties of directional entropy functions.

This research is supported in part by BSRI and KOSEF 95-0701-03-3.

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Authors and Affiliations 1. Department of Mathematics Ajou University Suwon Korea