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Continuity of directional entropy for a class of \(Z^ 2\)-actions. (English) Zbl 0847.28007

J. Milnor introduced the notion of directional entropy. Let \((X, T, {\mathcal F}, \mu)\) be a symbolic system and let \(S\) be an invertible measure preserving block map. Then \(S\) generates a \(\mathbb{Z}^2\) action on \(X\). Let \(P\) be a partition of \(X\). The directional entropy in the direction \({\mathbf v}\) is then \[ h({\mathbf v})= \sup_{B} \limsup_{t\to \infty} {1\over t} H\Biggl( \bigvee_{(i, j)\in B+ [0, t]{\mathbf v}} T^i S^j(P)\Biggr), \] where \(B\) is any bounded set in \(\mathbb{Z}^2\) and \[ B+ [0, t]{\mathbf v} = \{(i, j)\in \mathbb{Z}^2: \exists (k, \ell)\in B\text{ such that }(i, j)- (k, \ell) = \alpha{\mathbf v}\text{ for some } \alpha\in [0, t]\}. \] The author relates cone entropy to directional entropy under certain conditions. Since cone entropy is upper semicontinuous this allows him to prove upper semicontinuity for directional entropy for a wider class of maps than before.

MSC:

28D20 Entropy and other invariants
28D15 General groups of measure-preserving transformations
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