O.M. Karpel
Verkin Institute for Low Temperature Physics and Engineering of National Academy of Sciences of Ukraine, Kharkiv

INVARIANT MEASURES ON BRATTELI DIAGRAMS
Information from scientific report at the meeting of Presidium of NAS of Ukraine, June 17, 2015

Abstract:
The results, stated in the report, are devoted to the problem of classification of Cantor dynamical systems up to orbit equivalence. The study of invariant measures for such dynamical systems plays the crucial role in the classification. The complete classification of the wide class of both finite and infinite Borel measures on Cantor spaces is given. In particular, the complete classification of ergodic invariant measures for aperiodic substitution dynamical systems is obtained. Such measures are invariant for the cofinal equivalence relation on the path spaces of stationary Bratteli diagrams. The structure of ergodic measures on the path space of an arbitrary Bratteli diagram is studied, in particular, the description of subdiagrams supporting finite ergodic invariant measures is found. These results are essential for deciding an open question about the classification of Cantor aperiodic dynamical systems up to orbit equivalence.
Keywords: Cantor dynamical system, ergodic invariant measure, Bratteli diagram, orbit equivalence.

Language of article: ukrainian

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