Karpel O. Measures on Cantor sets and their classification

Purpose of the work is the classification of Borel measures on Cantor sets with respect to homeomorphisms of Cantor sets. The objects of research are Borel measures on Cantor sets and their supports, stationary Bratteli diagrams. The results obtained are new. The main results are the following. The necessary and sufficient conditions are found for the measures from the class of probability and infinite ergodic invariant measures for aperiodic substitution dynamical systems to be homeomorphic. The criterion is found for the measures from the class of infinite non-defective Borel measures on a compact Cantor space to be homeomorphic. The existence of continuum pairwise non-homeomorphic full non-atomic infinite Borel measures on a compact Cantor set is proved. The criterion is found for measures from the class of probability and the class of infinite Borel measures on a non-compact locally compact Cantor space to be homeomorphic.