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The relationship between word complexity and computational complexity in subshifts

The first author gratefully acknowledges the support of NSF grant DMS-1500685. The second author gratefully acknowledges the support of ANR grants ANR-12-BS02-0007 and 19-CE48-0007- 01
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  • We prove several results about the relationship between the word complexity function of a subshift and the set of Turing degrees of points of the subshift, which we call the Turing spectrum. Among other results, we show that a Turing spectrum can be realized via a subshift of linear complexity if and only if it consists of the union of a finite set and a finite number of cones, that a Turing spectrum can be realized via a subshift of exponential complexity (i.e. positive entropy) if and only if it contains a cone, and that every Turing spectrum which either contains degree $ {{\mathbf{0}}}$ or is a union of cones is realizable by subshifts with a wide range of 'intermediate' complexity growth rates between linear and exponential.

    Mathematics Subject Classification: Primary: 37B10; Secondary: 03D15, 03D25.

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