On directional entropy functions
Given a \({\mathbb Z}^2\)-process \((X,T,P)\), a directional entropy \(h({\mathbf v})\) is defined on the set of vectors \({\mathbf v}\in{\mathbb R}^2\) of unit length, by \[ h({\mathbf v},P)=\sup_{B}\limsup_{t\to\infty} (1/t)H\left(\bigvee_{{\mathbf n}\in B+[0,t]{\mathbf v}}T^{-{\mathbf n}}(P)\right), \] where the supremum is taken over all bounded subsets of \({\mathbb R}^2\). The main result here shows that the existence of an \(L\) for which \[ \sum_{n=0}^{\infty}H\left(T^{(0,-1)}(P)|\vee_{i=-n}^{n}\vee_{j=-L}^{0}T^{-(i,j)}(P)\right)<\infty \] is enough to ensure that \({\mathbf v}\mapsto h({\mathbf v})\) is continuous. In particular, this shows that a continuous \({\mathbb Z}^2\)-action on a compact metric space with an expansive direction has continuously varying directional entropy (answering a question raised by \textit{J. Milnor} [Complex Syst. 2, 357-385 (1988; Zbl 0672.68025)] in the setting of cellular automata).
scientific article; zbMATH DE number 1389710