Dynamical properties of some adic systems with arbitrary orderings

Sarah Frick, Karl Petersen, Sandi Shields

Research output: Contribution to journalArticle

Abstract

We consider arbitrary orderings of the edges entering each vertex of the (downward directed) Pascal graph. Each ordering determines an adic (Bratteli-Vershik) system, with a transformation that is defined on most of the space of infinite paths that begin at the root. We prove that for every ordering the coding of orbits according to the partition of the path space determined by the first three edges is essentially faithful, meaning that it is one-to-one on a set of paths that has full measure for every fully supported invariant probability measure. We also show that for every the subshift that arises from coding orbits according to the first edges is topologically weakly mixing. We give a necessary and sufficient condition for any adic system to be topologically conjugate to an odometer and use this condition to determine the probability that a random order on a fixed diagram, or a diagram constructed at random in some way, is topologically conjugate to an odometer. We also show that the closure of the union over all orderings of the subshifts arising from codings of the Pascal adic by the first edge has superpolynomial complexity, is not topologically transitive, and has no periodic points besides the two fixed points, while the intersection over all orderings consists of just four orbits.

LanguageEnglish (US)
Pages2131-2162
Number of pages32
JournalErgodic Theory and Dynamical Systems
Volume37
Issue number7
DOIs
StatePublished - Oct 1 2017

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Subshift
Pascal
Orbits
Coding
Orbit
Arbitrary
Diagram
Path Space
Path
Periodic Points
Faithful
Invariant Measure
Probability Measure
Closure
Union
Intersection
Fixed point
Partition
Roots
Necessary Conditions

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Dynamical properties of some adic systems with arbitrary orderings. / Frick, Sarah; Petersen, Karl; Shields, Sandi.

In: Ergodic Theory and Dynamical Systems, Vol. 37, No. 7, 01.10.2017, p. 2131-2162.

Research output: Contribution to journalArticle

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