REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS
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概要
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We consider representations of Cuntz algebras on self-similar fractal sets for proper/improper systems of contractions. Natural representations, called Hausdorff representations, are associated with self-similar sets and Hausdorff measures in the case of similitudes in $ R^n $. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list$ (\lambda_1 ^D, \ldots ,\lambda_N ^D) $, where $ \lambda_j $ is the Lipschitz constant of the $ j $th contraction and $ D $ is the Hausdorff dimension of the fractal set. Any non-trivial list can be realized by similitudes on the unit interval. There exists an improper system of contractions such that its representation of a Cuntz algebra on the self-similar fractal set is not unitarily equivalent to any Hausdorff representation for a proper system of similitudes in $ R^n $.
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