Computing the critical dimensions of Bratteli–Vershik systems with multiple edges

Abstract

The critical dimension is an invariant that measures the growth rate of the sums of Radon–Nikodym derivatives for non-singular dynamical systems. We show that for Bratteli–Vershik systems with multiple edges, the critical dimension can be computed by a formula analogous to the Shannon–McMillan–Breiman theorem. This extends earlier results of Dooley and Mortiss on computing the critical dimensions for product and Markov odometers on infinite product spaces.

The critical dimension for -measures

“…In this section we follow [3,5] and compute the critical dimension using two variants on the law of large numbers. The first lemma follows in the footsteps of [5].…”

“…The asymptotic rate of growth of the sum n i=1 ω i (x), called the critical dimension, was first studied in [11]. The critical dimension was later shown to be computable by a formula similar to the Shannon-McMillan-Breiman theorem for product measures [5] and Markov measures on Bratteli-Vershik diagrams [3].…”