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 ℝn. We completely classify the Hausdorff representations up to unitary equivalence. The complete invariant is the list (λ 1D,..., λND), where λj is the Lipschitz constant of the jth 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 ℝn.
All Science Journal Classification (ASJC) codes
- General Mathematics