For any **adic** transformation T {\displaystyle T} defined on the path space X {\displaystyle X} of an ordered Bratteli diagram, endowed with a Markov measure μ {\displaystyle \mu } , we construct an explicit dimension space (which corresponds to a matrix values random walk on Z {\displaystyle \mathbb {Z} } ) whose Poisson boundary can be identified as a Z {\displaystyle \mathbb {Z} } -space with the dynamical system ( X , μ , T ) {\displaystyle (X,\mu ,T)} .

2015, Thierry Giordano, David Handelman, Radu B. Munteanu, “Nonsingular transformations and dimension spaces”, in arXiv