American Institute of Mathematical Sciences

January  2010, 26(1): 333-346. doi: 10.3934/dcds.2010.26.333

Projectional entropy and the electrical wire shift

 1 Centro de Modelamiento Matemático, Universidad de Chile, Av. Blanco Encalada 2120, Piso 7, Santiago de Chile

Received  January 2009 Revised  July 2009 Published  October 2009

In this paper we present an extendible, block gluing $\z^3$ shift of finite type $\w^{el}$ in which the topological entropy equals the $L$-projectional entropy for a two-dimensional sublattice $L$<$\z^3$, even so $\w^{el}$ is not a full $\z$ extension of $\w^{el}_L$. In particular this example shows that Theorem 4.1 of [4] does not generalize to $r$-dimensional sublattices $L$ for $r>1$.
Nevertheless we are able to reprove and extend the result about one-dimensional sublattices for general $\z^d$ shifts - instead of shifts of finite type - under the same mixing assumption as in [4] and by posing a stronger mixing condition we also obtain the corresponding statement for higher-dimensional sublattices.
Citation: Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333
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