Invariant measures for iterated function systems with inverses

Abstract

We consider iterated function systems that contain inverses in the overlapping case. We focus on the parameterized families of iterated function systems with inverses, satisfying the transversality condition. We show that the invariant measure is absolutely continuous for a.e. parameter when the random walk entropy is greater than the Lyapunov exponent. We also show that if the random walk entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value.