On connectedness and discontinuity of invariant sets (Q1358508)

Let \((X,d)\) be a complete metric space. A set \(M\subseteq X\) is called to be invariant with respect to a collection of contraction mappings \(L=\{s_1,\dots,s_N\}\) acting on \(X\) if \(M=\bigcup_{i=1}^Ns_iM\). \textit{J. E. Hutchinson} [Indiana Univ. Math. J. 30, No. 5, 713-747 (1981; Zbl 0598.28011)] has proved that for any finite collection \(L\) there exists a unique compact set invariant with respect to \(L\). We denote it by \(|L|\). For any contraction mapping \(s\) the Lipschitz constant is defined by \(\text{Lip}s= \sup_{x\neq y}d(s(x),s(y))/d(x,y)\leq1\). We say that the invariant set \(|L|\) is totally discontinuous if \(s_i|L|\cap s_j|L|=\emptyset\) for \(i\neq j\). The authors prove that if an integer \(N\) and points \(c_1,\dots,c_N\in X\) are given then there exists a number \(k\in(0,1)\) such that for any collection of contraction mappings \(L=\{s_1,\dots,s_N\}\) with \(s_i(c_i)=c_i\) and \(\text{Lip}s_i<k\) for all \(i=1\), \dots, \(N\), the invariant set \(|L|\) is totally discontinuous. If \(X\) is the Euclidean space \({\mathbb R}^n\) and each \(s_i\) is a nondegenerate linear contraction mapping then if \(\min_i\det s_i>1/N\), then the invariant set \(|L|\) is not totally discontinuous and if \(\min_i\det s_i>1/2\), then the invariant set \(|L|\) is arcwise connected.