REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS REPRESENTATIONS OF CUNTZ ALGEBRAS ON FRACTAL SETS

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 $.

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 <B>R</B><SUP><I>n</I></SUP>. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list(λ<SUB>1</SUB><SUP>D</SUP>, . . . ,λ<SUB>N</SUB><SUP>D</SUP>), where λ<SUB>j</SUB> is the Lipschitz constant of the <I>j</I> th contraction and <I>D</I> 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 <B>R</B><SUP><I>n</I></SUP>.