Abstract
Abstract
We prove results about subshifts with linear (word) complexity, meaning that
$\limsup \frac {p(n)}{n} < \infty $
, where for every n,
$p(n)$
is the number of n-letter words appearing in sequences in the subshift. Denoting this limsup by C, we show that when
$C < \frac {4}{3}$
, the subshift has discrete spectrum, that is, is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with
$C = \frac {3}{2}$
which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether
$C = \frac {5}{3}$
was the minimum possible among such subshifts; our results show that the infimum in fact lies in
$[\frac {4}{3}, \frac {3}{2}]$
. All results are consequences of a general S-adic/substitutive structure proved when
$C < \frac {4}{3}$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
1 articles.
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1. Multiplicity of topological systems;Ergodic Theory and Dynamical Systems;2024-02-05