Dynamical properties of some adic systems with arbitrary orderings

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 $k$ the subshift that arises from coding orbits according to the first $k$ 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

Periodic codings of Bratteli-Vershik systems

“…In addition, in the cutting and stacking presentation the base of the tower does not come from a spacer reservoir. We then show that within this class some k-factor is finite if and only if eventually all levels of the diagram satisfy what we call the local deficit condition with respect to level k. This condition, which extends the definition of uniformly ordered in [19], imposes a structure on the partial ordering of the edges and how the spacers are interspersed, producing a sort of local incomplete periodicity (see Definitions 4.10 and 4.1). Section 5 presents examples of the various possibilities for k-factors.…”

“…In this section we consider more general Bratteli-Vershik systems, still satisfying the conditions (C1)-(C4) in Section 2. We begin with the following definition that will help us find conditions for k-codings to be periodic and for the systems to be isomorphic or topologically conjugate to odometers, generalizing results in [19]. As mentioned in the Introduction and discussed more fully below, this definition generalizes the idea of uniformly ordered in [19] to a sort of incomplete periodicity.…”

“…Remark 4.7. The semi k-periodic property at level k + 1 is a generalization of uniformly ordered at level k+1 that appears in [19]. The latter is equivalent to the existence of a block W k+1 ∈ A * k such that for each j = 1, .…”

“…Our main goal is to determine when a system is essentially k-expansive for some k. For the Pascal system with the usual left-right ordering of incoming edges, essential 1-expansiveness was proved in [28]; see also [30] and a similar result for classes of systems in [29,20]. Essential 3-expansiveness for the Pascal system with any ordering of the edges was proved in [19]. [11] showed that the topological dynamical system defined by a simple, properly ordered Bratteli-Vershik diagram with a bounded number of vertices per level, which they called topological finite rank but which we prefer to call bounded width, is either (topologically) expansive (meaning that for some k the partition α k generates the topology) or else the system is topologically conjugate to an odometer.…”

“…[11] showed that the topological dynamical system defined by a simple, properly ordered Bratteli-Vershik diagram with a bounded number of vertices per level, which they called topological finite rank but which we prefer to call bounded width, is either (topologically) expansive (meaning that for some k the partition α k generates the topology) or else the system is topologically conjugate to an odometer. In [19] we gave necessary and sufficient conditions for a simple properly ordered Bratteli-Vershik system to be topologically conjugate to an odometer: there should be infinitely many uniformly ordered levels. Below we extend this condition to more general diagrams.…”