Function dynamics (Q5944164)

The asymptotical dynamical behaviour of the sequence of maps \((f_n)_{n=0}^\infty\) is studied, where \(f_0\) is a continuous map from the interval \([0,1]\) into itself and \(f_{n+1}=g_n\circ f_n\), with \(g_n(x)=(1-\varepsilon)x+\varepsilon f_n(x)\) for a fixed \(0<\varepsilon<1\). For the (non-empty) set \(\Omega\) of points where it makes sense, let \(f_\infty\) be defined by \(f_\infty(x)=\lim_{n\rightarrow \infty} f_n(x)\). Also, if we denote \(\Omega_q=\{x\in \Omega: f_\infty(x)=q\}\) for any fixed point \(q\) of \(f_\infty\) (when \(\Omega=\bigcup_q \Omega_q\)), we define \(g_\infty\) on \(\Omega\) by \(g_\infty(x)=(1-\varepsilon)x+\varepsilon q\) whenever \(x\in \Omega_q\). It is shown that the maps \(f_\infty\) and \(g_\infty\) play a prominent role in the description of this dynamics which, as illustrated with examples, may be rather simpler than expected (including the appearance of hierarchical and self-referential structures).