While it remains open if every subshift over a finite alphabet has a characteristic measure, we define a new class of shifts, which we call language stable subshifts, and show that these shifts have characteristic measures. Frisch and Tamuz recently dubbed such measures characteristic, and further showed that every zero entropy subshift has a characteristic measure. It follows readily from a classical result of Parry that the full shift on finitely many symbols, and more generally any mixing subshift of finite type, supports such a measure. This is a large class that is generic in several senses and contains numerous positive entropy examples.ĪB - We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. N2 - We consider the problem of when a symbolic dynamical system supports a Borel probability measure that is invariant under every element of its automorphism group. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature. The second author was partially supported by NSF grant DMS-1800544. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.T1 - Characteristic measures for language stable subshifts To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. We use a disjointness argument and the key ingredient in our proof is a structural result for measure preserving systems naturally associated with the M\"obius and the Liouville function. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. The M\"obius disjointness conjecture of Sarnak states that the M\"obius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. In the particular case of topological rank 2 subshifts, we prove their complexity is always subquadratic along a subsequence and their automorphism group is trivial. We also exhibit that finite topological rank does not imply non-superlinear complexity. We establish that such systems, when they are expansive, define the same class of systems, up to topological conjugacy, as primitive and recognizable $$-adic subshifts. Minimal Cantor systems of finite topological rank (that can be represented by a Bratteli-Vershik diagram with a uniformly bounded number of vertices per level) are known to have dynamical rigidity properties.
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