Continuity of the Hausdorff Dimension for Sub - Self - Conformal Sets ( Communicated
نویسندگان
چکیده
Self-similar sets and self-conformal sets have been studied extensively. Recently, Falconer introduced sub-self-similar sets for a generalization of self-similar sets, and obtained the Hausdorff dimension and Box dimension for these sets if the open set condition (OSC) is satisfied. Chen and Xiong proved the continuity of the Hausdorff dimension for sub-self-similar sets under the assumption that the self-similar iterated function system (IFS) satisfies the OSC[12]. We extend the notion of sub-self-similar sets to sub-self-conformal sets. In this paper, we study the continuity of the Hausdorff dimension for sub-self-conformal sets. For self-conformal sets some well-known properties of self-similar sets are not true in general. So Chen and Xiong’s method does not work in the case of sub-self-conformal sets. We offer a method to deal with it at first. And then, by using the property of the shift invariant set in symbolic space we prove the continuity of the Hausdorff dimension for sub-self-conformal sets.
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