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\iteman{io-port 05582805}
\itemau{Madden, K.M.}
\itemti{A single nonexpansive, nonperiodic rational direction.}
\itemso{Complex Syst. 12, No. 2, 253-260 (2000).}
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Summary: A two-dimensional cellular automaton consists of a two-dimensional lattice of sites, each of which takes on a finite number of values, and a cellular automaton map. The cellular automaton map updates the value at each site $a\in\Bbb Z^2$ using a translation invariant rule that only depends on the values at the sites in some finite neighborhood of $a$. A number of global properties of a two-dimensional cellular automata, such as the directional entropies introduced by Milnor, can be studied using the methods of dynamical systems. In this work we consider $E$, the expansive one-dimensional subspaces of $\Bbb R^2$ as defined by {\it M. Boyle} and {\it D. Lind} [Trans. Am. Math. Soc. 349, No. 1, 55--102 (1997; Zbl 0863.54034)]. Various properties of cellular automata, including Milnor's directional entropies, vary nicely within connected components of $E$ so it is natural to ask what subsets of $\Bbb R^2$ may occur as expansive one-dimensional subspaces. Boyle and Lind give an almost complete answer, the single unresolved case being when $E$ is the complement of a line with irrational slope. In this work we construct a related example with the potential to shed light on the unresolved case.
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