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Measure and dimension of solenoidal attractors of one dimensional dynamical systems

Authors:	A M Blokh M Yu Lyubich

Institution:	(1) Steklov Mathematical Institute, LOMI, Fontanka 27, SU-191011 Leningrad, USSR

Abstract:	Letf:MM be aC -map of the interval or the circle with non-flat critical points. A closed invariant subsetAM is called a solenoidal attractor off if it has the following structure: $A\mathop \cap \limits_{n = 1}^\infty \mathop \cup \limits_{k = 0}^{P_n - 1} l_k^{(n)}$ , where{I _k ⁽ⁿ⁾ is the cycle of intervals of periodp _n. We prove that the Lebesgue measure ofA is equal to zero and if sup(p _n+1/p_n)< then the Hausdorff dimension ofA is strictly less than 1.

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