Let f:M→M be a C∞-map of the interval or the circle with non-flat critical points. A closed invariant subset A⊂M is called a solenoidal attractor of f if it has the following structure: {Mathematical expression}, where {Ik(n) is the cycle of intervals of period pn→∞. We prove that the Lebesgue measure of A is equal to zero and if sup(pn+1/pn)<∞ then the Hausdorff dimension of A is strictly less than 1. © 1990 Springer-Verlag.