Structural instability of cookie-cutter sets (Q881625)

An invariant set \(E\) of the IFS \(\{f_i\}\) is said to be structurally instable in \(C^1\) topology if one can always perturb \(\{f_i\}\) arbitrarily small in \(C^1\) topology to provide an IFS \(\{g_i\}\) with its invariant set \(F\) such that \(\dim E = \dim F\), but \(E\) and \(F\) are not Lipschitz equivalent. In this paper the author proves that the cookie-cutter set in \({\mathbb R}\) is structurally instable in \(C^1\) topology. For proving this theorem, he considers a homotopic deformation between two IFSs and uses the continuity of Hausdorff dimensions in \(C^1\) topology proved by himself [J. Lond. Math. Soc. (2) 70, No. 2, 369--382 (2004; Zbl 1160.28305)].