×

On stochastic stability of non-uniformly expanding interval maps. (English) Zbl 1290.37022

Let \(f:[0,1]\to [0,1]\) be \(C^3\) multimodal interval map with no attracting cycles and non-flat critical points. Further, assume that the function satisfies a summability condition of exponent one, i.e., \(\sum_{n=0}^\infty \frac{1}{|D^n(f(v))|}<\infty\) for any critical value \(v\). Choose independent random maps \(g_n\) from a space \(\Omega\) containing \(f\) which may be in some case a small neighbourhood of \(f\) in the \(C^2\) topology. Then under general conditions on the perturbations, a typical random orbit \(g_n\circ ...\circ g_0(x)\) has “roughly the same distribution as a typical orbit of \(f\)” or more precisely, strong stochastic stability is shown.

MSC:

37E05 Dynamical systems involving maps of the interval
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37C75 Stability theory for smooth dynamical systems
37H99 Random dynamical systems
PDFBibTeX XMLCite
Full Text: DOI arXiv