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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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: A trajectory over the finite field $ {F_q}$ is a function from the integers I to $ {F_q}$. The set $ X({F_q})$ of all trajectories over $ {F_q}$ is a topological vector space in the product topology induced by the discrete topology on $ {F_q}$, and coordinatewise addition and scalar multiplication of trajectories. Let $ \phi $ be a continuous linear operator on $ X({F_q})$ which commutes with the shift. If x is a trajectory over $ {F_q}$, then the $ \phi $-orbit of x is the sequence of trajectories $ x,\phi (x),{\phi ^2}(x), \cdots $. Suppose that $ \phi $ is not a scalar multiple of the identity. Theorem. The trajectory x is periodic if and only if the $ \phi $-orbit of x is eventually periodic.


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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