Let $S$ be the adic transformation preserving a probability measure $\mu$ on the set $\Gamma$ of infinite paths of a $\mathbb{N}$-graded ordered Bratteli graph.
For every $n \geq 0$ define the equivalence relation ${\cal R}_n$ on $\Gamma$ by $$ \gamma {\cal R}_n \gamma' \,\iff\, \gamma_k=\gamma'_k \quad\text{for every k $\geq$ n}. $$ There are $d_n(\gamma)$ paths in the ${\cal R}_n$-equivalence class ${\cal R}_n(\gamma)$ of a path $\gamma$, where $d_n(\gamma)$ is the "dimension" of the vertex at level $n$ through which passes $\gamma$ (the "dimension" of a vertex is the number of paths from this vertex to the root vertex at level $0$). They are ordered: there is a minimal path among them, say $\bar\gamma_n$, and one has $$ {\cal R}_n(\gamma) = \{\bar\gamma_n, S\bar\gamma_n, \ldots, S^{d_n(\gamma)-1}\bar\gamma_n\}. $$ Thus there is an integer $k_n(\gamma)$ such that $\bar\gamma_n = S^{-k_n(\gamma)}\gamma$ and $$ {\cal R}_n(\gamma) = \{S^{-k_n(\gamma)}\gamma, S^{-k_n(\gamma)+1}\gamma, \ldots, S^{-k_n(\gamma)+d_n(\gamma)-1}\gamma\}. $$ It seems true to me that $\boxed{k_n(\gamma) \to \infty}$ and $\boxed{-k_n(\gamma)+d_n(\gamma) \to \infty}$ for $\mu$-almost all $\gamma$, but I don't see how to prove it. How could one justify this is indeed true ?
