The critical dimension for measurable dynamics on Bratteli-Vershik diagrams DANIEL F. MANSFIELD†, ANTHONY H. DOOLEY‡ † School of Mathematics and Statistics, University of New South Wales, Sydney, Australia (e-mail: daniel.mansfield@unsw.edu.au) ‡ Department of Mathematical Sciences, University of Bath, Bath, United Kingdom (e-mail: a.h.dooley@bath.ac.uk) (Received 17 October 2013) Abstract. The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon-Nikodym derivatives. This has been shown to equal the average coordinate entropy for product odometers and for Markov odometers on Bratelli-Vershik systems with bounds on the numbers of edges. We simplify and extend the proofs to a larger class of systems where the numbers of edges may grow exponentially for product odometers, and sub- exponentially for Markov odometers. 1. Introduction Bratteli introduced the use of certain infinite graphs as a model for the topology of non-singular dynamical systems for the study of C ∗ algebras. Vershik introduced the adic transformation on the path space of a Bratteli diagram to study AF- algebras. Together this structure is called a Bratteli-Vershik diagram[8]. Dooley and Hamachi showed that every non-singular dynamical system is orbit equivalent to a Markov odometer on a Bratteli-Vershik diagram. Hence we can work with measurable dynamics instead of topological dynamics. This result can be combined with the work of Krieger to conclude that every amenable factor of a separable von Neumann algebra can be generated by the group measure space construction from a Markov odometer on a Bratteli-Vershik diagram. The additional structure opens the possibility of saying more about orbit equivalence classes using the properties of a Markov odometer. One such property is the critical dimension. Mortiss and Dooley [9, 2], defined the upper and lower critical dimension for a non-singular dynamical system. They proved that for a product odometer with a bounded number of edges at each level   E (n)   <C , the upper and lower critical dimension can be computed by a formula analogous to