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Parabolic Cantor sets. (English) Zbl 0895.58036

The theory of Cantor repellors is quite well developed both in the setting of conformal dynamics and for smooth maps on the real line. Examples of questions that have been studied include their ergodic theory, conformal measures, classification up to Hölder or Lipschitz-continuous conjugacies and the phenomena of rigidity where a Lipschitz-continuous equivalence already implies analytic equivalence.
The present paper considers more general invariant Cantor sets which can contain fixed parabolic points. For these “parabolic Cantor sets” the same questions as in the case of repellers are studied. The results are generally similar. The proofs follow the author’s technique used in other papers on systems with parabolic points.

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37E99 Low-dimensional dynamical systems
37A99 Ergodic theory
28A80 Fractals
28A78 Hausdorff and packing measures
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