Abstract
Letf:M→M be aC ∞-map of the interval or the circle with non-flat critical points. A closed invariant subsetA⊂M 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 (n)k 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/pn)<∞ then the Hausdorff dimension ofA is strictly less than 1.
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Communicated by Ya. G. Sinai
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Blokh, A.M., Lyubich, M.Y. Measure and dimension of solenoidal attractors of one dimensional dynamical systems. Commun.Math. Phys. 127, 573–583 (1990). https://doi.org/10.1007/BF02104502
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DOI: https://doi.org/10.1007/BF02104502