## 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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### References:

 [1] Bowen, Publ. IHES 50 pp 11– (1980) · Zbl 0439.30032 [2] DOI: 10.1007/BF01762666 · Zbl 0299.54031 [3] Abramov, Amer. Math. Soc. Translations 49 pp 167– (1966) · Zbl 0185.21803 [4] Williams, Publ. IHES 50 pp 73– (1979) · Zbl 0484.58021 [5] Urba?ski, Ergod. Th. & Dynam. Sys. 6 pp none– (1986) [6] Misiurewicz, Studia Math. 67 pp 45– (1980) [7] DOI: 10.1007/BF02760884 · Zbl 0422.28015 [8] McCluskey, Ergod. Th. & Dynam. Sys. 3 pp 251– (1983) [9] DOI: 10.1007/BF01223133 · Zbl 0399.28011 [10] DOI: 10.1007/BF01215004 · Zbl 0485.28016 [11] Kuratowski, Topology (1966) [12] DOI: 10.1007/BF02761854 · Zbl 0456.28006 [13] Misiurewicz, Bull. Acad. Pol. Sci. 24 pp 1069– (1976)
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