Realizing ergodic properties in zero entropy subshifts
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REALIZING ERGODIC PROPERTIES IN ZERO ENTROPY SUBSHIFTS
BY
Van Cyr Department of Mathematics, Bucknell University, Lewisburg, PA 17837, USA e-mail: [email protected]
AND
Bryna Kra∗ Mathematics Department, Northwestern University, Evanston, IL 60208, USA e-mail: [email protected]
ABSTRACT
A subshift with linear block complexity has at most countably many ergodic measures, and we continue the study of the relation between such complexity and the invariant measures. By constructing minimal subshifts whose block complexity is arbitrarily close to linear but have uncountably many ergodic measures, we show that this behavior fails as soon as the block complexity is superlinear. With a different construction, we show that there exists a minimal subshift with an ergodic measure such that the lim inf of the slow entropy grows slower than any given rate tending to infinitely but the lim sup grows faster than any other rate majorizing this one yet still growing subexponentially. These constructions lead to obstructions in using subshifts in applications to properties of the prime numbers and in finding a measurable version of the complexity gap that arises for shifts of sublinear complexity.
∗ The second author was partially supported by NSF grant 1800544.
Received March 4, 2019 and in revised form September 9, 2019
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V. CYR AND B. KRA
Isr. J. Math.
1. Introduction Assume that (X, σ) is a subshift over the finite alphabet A, meaning that X ⊂ AZ is a closed set that is invariant under the left shift σ : AZ → AZ . The block complexity pX (n) of the shift is defined to be the number of words of length n which occur in any x ∈ X. Boshernitzan [1] showed that a minimal subshift with linear block complexity has only finitely many ergodic measures, where the number depends on the complexity growth. In [2], we showed that any subshift (minimal or not) with linear block complexity has at most finitely many nonatomic ergodic measures, and so at most countably many ergodic measures (with no requirement that the measures are nonatomic). In the same article, we give examples of subshifts with block complexity arbitrarily close to linear which have countably many nonatomic ergodic measures. Our main result is to show there is no complexity bound beyond linear on a subshift that suffices for guaranteeing there are at most countably many ergodic measures. More precisely, we show as soon as the growth is superlinear, we can have the maximal number of ergodic measures: Theorem 1.1: If (pn )n∈N is a sequence of natural numbers such that pn = ∞, lim inf n→∞ n then there exists a minimal subshift X which supports uncountably many ergodic measures and is such that lim inf n→∞
PX (n) = 0. pn
The distinction between countably and uncountably many ergodic measures supported by a subshift has recently received attention, as it plays a role in the deep results of Frantzikinakis and Host [4] on the complexity of the Liouville shift. More precisely, by studying the subshift naturally associated to Liouville function λ(n) (see Section 2.4
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