Invariant measures on stationary Bratteli diagrams

Abstract

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

Quasimorphismes sur le monoïde libre, et substitutions dans les mesures invariantes

“…Cette formule définit une fonction sur les boréliens de B N , et on voit facilement qu'elle est σ-additive ; c'est donc bien une mesure sur B N . Par construction, elle vérifie la condition (3.8) pour les fonctions caractéristiques d'ensembles mesurables, et par linéarité, pour toutes les fonctions f étagées (3) . Le cas général d'une fonction mesurable bornée f : B N → R s'en déduit, en notant que toute fonction mesurable bornée est limite uniforme de fonctions étagées (4) .…”

Nonstationary Mixing and the Unique Ergodicity of Adic Transformations

“…We have mostly considered primitive sequences or, at worst, sequences which are primitive after being reduced; moving beyond primitivity is much of the motivation behind [BKMS09]. An important example with a nonprimitive matrix sequence is the Pascal adic transformation; there, as shown by Vershik, one has a continuum of invariant probability measures, and the same holds for the cutting and stacking example of [HIK72] which is modelled by the Pacal adic.…”

“…This argument can be extended to the general primitive as well as certain nonprimitive cases, including the Chacon example (see §5.1) and the infinite measure example of [Fis92], see in particular Remark 2.10 of [BKMS09] regarding the infinite measure case. Their proof is related to the column-stochastic method just mentioned, as both can be understood via a cuttingand-stacking construction of the invariant measures.…”

“…After this paper and [FFT09] had been submitted, we received the preprint [BKMS09]; despite its title and abstract, in fact the nonstationary case is also addressed, see Theorem 2.8 there, and in particular the authors have independently come up with a completely di↵erent proof from that of this paper that the topological PerronFrobenius property implies unique ergodicity. This argument can be extended to the general primitive as well as certain nonprimitive cases, including the Chacon example (see §5.1) and the infinite measure example of [Fis92], see in particular Remark 2.10 of [BKMS09] regarding the infinite measure case.…”

Counting paths in Bratteli diagrams for SU(2) k

“…For our scope, it is sufficient to give here an "algebra free" description of this notion (see, e.g., [1] or [3]). More specifically, lattice points will be labeled by numbers instead of Young diagrams.…”