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arXiv · 1709.00055

Exact number of ergodic invariant measures for Bratteli diagrams

S. Bezuglyi·O. Karpel·J. Kwiatkowski
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Abstract

For a Bratteli diagram $B$, we study the simplex $\mathcal{M}_1(B)$ of probability measures on the path space of $B$ which are invariant with respect to the tail equivalence relation. Equivalently, $\mathcal{M}_1(B)$ is formed by probability measures invariant with respect to a homeomorphism of a Cantor set. We study relations between the number of ergodic measures from $\mathcal{M}_1(B)$ and the structure and properties of the diagram $B$. We prove a criterion and find sufficient conditions of unique ergodicity of a Bratteli diagram, in which case the simplex $\mathcal{M}_1(B)$ is a singleton. For a finite rank $k$ Bratteli diagram $B$ having exactly $l \leq k$ ergodic invariant measures, we explicitly describe the structure of the diagram and find the subdiagrams which support these measures. We find sufficient conditions under which: (i) a Bratteli diagram has a prescribed number (finite or infinite) of ergodic invariant measures, and (ii) the extension of a measure from a uniquely ergodic subdiagram gives a finite ergodic invariant measure. Several examples, including stationary Bratteli diagrams, Pascal-Bratteli diagrams, and Toeplitz flows, are considered.

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BibTeXRIS

S. Bezuglyi, O. Karpel, J. Kwiatkowski. 2019-04-21. Exact number of ergodic invariant measures for Bratteli diagrams. https://arxiv.org/abs/1709.00055

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