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On the genesis of symbolic dynamics as we know it

Ethan M. CovenZbigniew H. Nitecki — 2008

Colloquium Mathematicae

We trace the beginning of symbolic dynamics-the study of the shift dynamical system-as it arose from the use of coding to study recurrence and transitivity of geodesics. It is our assertion that neither Hadamard's 1898 paper, nor the Morse-Hedlund papers of 1938 and 1940, which are normally cited as the first instances of symbolic dynamics, truly present the abstract point of view associated with the subject today. Based in part on the evidence of a 1941 letter from Hedlund to Morse, we place the...

Embedding odometers in cellular automata

Ethan M. CovenReem Yassawi — 2009

Fundamenta Mathematicae

We consider the problem of embedding odometers in one-dimensional cellular automata. We show that (1) every odometer can be embedded in a gliders-with-reflecting-walls cellular automaton, which one depending on the odometer, and (2) an odometer can be embedded in a cellular automaton with local rule x i x i + x i + 1 m o d n (i ∈ ℤ), where n depends on the odometer, if and only if it is “finitary.”

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