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.