Subdiagrams and invariant measures of Bratteli diagrams

We study ergodic measures on the path space $X_B$ of a Bratteli diagram $B$ invariant with respect to the tail equivalence relation $\mathcal E$. Our aim is to characterize those subdiagrams that support an ergodic finite invariant measure. Given a subdiagram $\overline B$ of a Bratteli diagram $B$, consider an ergodic probability measure $\nu$ on $X_{\overline B}$. This measure can be naturally extended (by $\mathcal E$-invariance) to a measure $\widehat{\nu}$ defined on the $\mathcal E$-saturation $\widehat{X}_{\overline{B}}$ of the path space $X_{\overline{B}}$. We give criteria and sufficient conditions for the finiteness of the extended measure. On the other hand, suppose that a probability measure $\mu$ is given on a diagram $B$. We answer the question when the path space of the subdiagram $\overline B$ has positive measure. The talk is based on the joint work with M. Adamska, S. Bezuglyi and J. Kwiatkowski.

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