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Invariant subsets of expanding mappings of the circle. (English) Zbl 0653.58031
Let f: \(S^ 1\to S^ 1\) be an expanding map of the circle. The author investigates entropy and geometric properties of invariant subsets \({\mathcal L}\subset {\mathcal S}^ 1\). Let us state one of the results. Let \({\mathcal K}_ m\) be the class of subsets \({\mathcal L}\) of the form \(L=\{x| \quad f^ nx\not\in {\mathcal U},\quad n=0,1,...\}\) where \({\mathcal U}\) is the union of m open intervals. Theorem 3. The Hausdorff dimension \({\mathcal H}{\mathcal D}({\mathcal L})\) continuously depends on \({\mathcal L}\in {\mathcal K}_ m\). Note that \({\mathcal H}{\mathcal D}({\mathcal L})\) is not a continuous functional on the space \({\mathcal K}\) of all invariant subsets \({\mathcal L}\subset {\mathcal S}^ 1\) since finite subsets are dense in \({\mathcal K}\).
Reviewer: M.Lyubich

MSC:
37D99 Dynamical systems with hyperbolic behavior
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