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Sofic subshifts and piecewise isometric systems

Published online by Cambridge University Press:  01 December 1999

AREK GOETZ
Affiliation:
Department of Mathematics, Boston University, 111 Cummington Street, Boston, MA 02215, USA (e-mail: [email protected]) Current address: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA.

Abstract

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

Type
Research Article
Copyright
1999 Cambridge University Press

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