Shift dynamical systems over finite fields

Author:
Melvyn B. Nathanson

Journal:
Proc. Amer. Math. Soc. **34** (1972), 591-594

MSC:
Primary 54H20

DOI:
https://doi.org/10.1090/S0002-9939-1972-0295321-2

MathSciNet review:
0295321

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Abstract | References | Similar Articles | Additional Information

Abstract: A trajectory over the finite field is a function from the integers *I* to . The set of all trajectories over is a topological vector space in the product topology induced by the discrete topology on , and coordinatewise addition and scalar multiplication of trajectories. Let be a continuous linear operator on which commutes with the shift. If *x* is a trajectory over , then the -orbit of *x* is the sequence of trajectories . Suppose that is not a scalar multiple of the identity. Theorem. *The trajectory x is periodic if and only if the* -*orbit of x is eventually periodic*.

**[1]**G. A. Hedlund,*Endomorphisms and automorphisms of the shift dynamical system*, Math. Systems Theory**3**(1969), 320–375. MR**0259881**, https://doi.org/10.1007/BF01691062

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0295321-2

Keywords:
Shift dynamical systems,
symbolic flows,
periodic trajectories,
periodic orbits,
sequence spaces

Article copyright:
© Copyright 1972
American Mathematical Society