Stability versus hyperbolicity in dynamical and iterated function systems (Q1396790)

Let \(K\) be a nonempty compact subset of the plane \({\mathbb R}^2\) with the ordinary Euclidean metric \(d\) and \(\mathcal F\) be a family of functions \(f\colon K\to K\). A pair \(\{\mathcal F,K\}\) is referred as an iterated function system (IFS). The IFS \(\{\mathcal F,K\}\) is hyperbolic if there exists a \(\lambda <1\) such that \(d(f(x),f(y))\leq \lambda d(x,y),\) for all \(f\in \mathcal F\) and all points \(x,y\in K\). Let \(\mathcal F^{\infty }\) denote the family of infinite sequences \(\{f_n\}_{n=1}^{\infty }\) of functions in \(\mathcal F\). If \(F=\{f_n\}_{n=1}^{\infty }\) is an element in \(\mathcal F^{\infty }\), then for each positive integer \(n\) the composite function \(F_n\) is given by \[ F_n=f_1\circ f_2\circ \dots \circ f_n=f_1(f_2(\dots (f_n)\dots)). \] The IFS \(\{\mathcal F,K\}\) is stable if, for every element \(F\in \mathcal F^{\infty }\), it holds diam\((F_n(K))\to 0\) as \(n\to \infty \). An IFS \(\{\mathcal F,K\}\) is conjugate to an IFS \(\{\mathcal F',K'\}\) if there exists a homeomorphism \(h\colon K\to K'\) such that \[ \mathcal F'=h\circ \mathcal F\circ h^{-1}=\{h\circ f\circ h^{-1}\colon f\in \mathcal F\}. \] The main result of the paper (Theorem 2.1) states that there exists a stable IFS consisting of a single function that is not conjugate to any hyperbolic IFS.