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Full-Text Articles in Physical Sciences and Mathematics
Measure-Theoretically Mixing Subshifts With Low Complexity, Darren Creutz, Ronnie Pavlov, Shaun Rodock
Measure-Theoretically Mixing Subshifts With Low Complexity, Darren Creutz, Ronnie Pavlov, Shaun Rodock
Mathematics: Faculty Scholarship
We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any f : N → N with f (n)/n increasing and ∑ 1/f (n) < ∞, that there exists an extremely elevated staircase with word complexity p(n) = o(f (n)). This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.
Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding
Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding
Mathematics: Faculty Scholarship
We prove that for any transitive subshift X with word complexity function cn(X), if lim inf(log(cn(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on cn(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be …
Subsystems Of Transitive Subshifts With Linear Complexity, Andrew Dykstra, Nicholas Ormes, Ronnie Pavlov
Subsystems Of Transitive Subshifts With Linear Complexity, Andrew Dykstra, Nicholas Ormes, Ronnie Pavlov
Mathematics: Faculty Scholarship
We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra …
On Subshifts With Slow Forbidden Word Growth, Ronnie Pavlov
On Subshifts With Slow Forbidden Word Growth, Ronnie Pavlov
Mathematics: Faculty Scholarship
In this work, we treat subshifts, defined in terms of an alphabet A and (usually infinite) forbidden list F, where the number of n-letter words in F has ‘slow growth rate’ in n. We show that such subshifts are well behaved in several ways; for instance, they are boundedly supermultiplicative in the sense of Baker and Ghenciu [Dynamical properties of S-gap shifts and other shift spaces. J. Math. Anal. Appl.430(2) (2015), 633–647] and they have unique measures of maximal entropy with the K-property and which satisfy Gibbs bounds on large (measure-theoretically) sets. The main tool in our proofs is a …