Absolutely continuous invariant measure of a map from grazing-impact oscillators

•We investigate a one-dimensional map with unbounded derivative.•The map is the limit of the Nordmark map which is the normal form of the impact oscillators near grazing states.•We prove that the map has an ergodic absolutely continuous invariant probability measure by constructing an induced Markov map.

In this paper we investigate a one-dimensional map with unbounded derivative. The map is the limit of the Nordmark map which is the normal form of a discrete time representation of impact oscillators near grazing, i.e. when the dissipation of the systems is large, the Nordmark map can be viewed as a perturbation of the one-dimensional map. We prove that the map has an ergodic absolutely continuous invariant probability measure in a region of parameter space by constructing an induced Markov map.