A class of iterated function systems with adapted piecewise constant transition probabilities: Asymptotic stability and Hausdorff dimension of the invariant measure

Abstract

A class of iterated function systems (IFS) with non-overlapping or just-touching contractions on closed real intervals and adapted piecewise constant transition probabilities are studied. We give criteria for the existence and the uniqueness of an invariant probability measure for the IFSs and for the asymptotic stability of the system in terms of bounds of transition probabilities. The proofs are mainly based on the symbolic system associated with the contractions, an extended alphabet and usual theorems for Markov chains. Additionally, in case there exists a unique invariant measure, we obtain its Hausdorff dimension as the ratio of the entropy over the Lyapunov exponent. This result extends the formula, established in the literature for continuous transition probabilities, to the case considered here of piecewise constant probabilities. The main idea of this proof consists in finding good lower and upper bounds for the invariant measure.