A contractivity condition for iterated function systems
This paper considers the convergence of measures in one-dimensional iterated function systems which are not uniformly contractive. The goal is to find conditions on the pointwise contractivity which guarantee long-term uniform contractivity. The system is said to be weakly contractive if \[ \prod_{k=1}^m |f_k'(x)|^{p_k}\leq r<1, \qquad \forall x\in X, \] where the system chooses the functions \(f_1,f_2,\dots,f_k\) with probabilities \(p_1,p_2,\dots,p_k\). This, together with some regularity conditions, implies convergence to a unique stationary distribution. A practicable sufficient condition for weak contractivity is given in terms of an arbitrary weight function \(w\), bounded away from 0 and \(\infty\): \[ \prod_{k=1}^m \left|\frac{f_k'(x)w(x)}{w(f_k(x))}\right|^{p_k}\leq r<1, \qquad \forall x\in X. \] This covers some ground missed in the similar results of \textit{P. Diaconis} and \textit{D. Freedman} [SIAM Rev. 41, No. 1, 45-76 (1999; Zbl 0926.60056)] and \textit{J. H. Elton} [Ergodic Theory Dyn. Syst. 7, 481-488 (1987; Zbl 0621.60039)]. It is also shown that convergence to a unique stationary distribution follows with no additional assumptions from the condition \[ \sum_{k=1}^m p_k\left|\frac{f_k'(x)w(x)}{w(f_k(x))}\right|\leq r<1, \qquad \forall x\in X. \] This is a special case of the condition given earlier. Applications are given to two population models, the logistic and the Ricker ones. Most of the results for logistic maps have been anticipated by the reviewer [Indag. Math., New Ser. 12, No. 4, 557-584 (2001)], \textit{Klünger} [``Stochastic population models'' (to appear)], and \textit{K. B. Athreya} and \textit{J. Dai} [J. Theor. Probab. 13, No. 2, 595-608 (2000; Zbl 0969.60069)]. An exception is a novel idea for finding good weight functions for parametrized systems concentrated in the range where a single fixed point has bifurcated, and is on its way to a cycle of period 2. This proves convergence in certain ranges of parameters for the logistic and Ricker systems which were inaccessible to earlier methods.
scientific article; zbMATH DE number 1892240