Implicit Equilibrium Dynamics∗ Alfredo Medio∗∗ and Brian Raines∗∗∗ ∗∗ Department of Statistics, University of Udine, Via Treppo 18, 33100, Udine, Italy ∗∗∗ Department of Mathematics, Baylor University, Waco, Texas, USA ∗ The authors would like to thank the Associate Editor of MD and two anonymous referees for many comments and criticisms which helped us correct several errors and significantly improve the quality of the paper. We are also grateful to Artur Avila and Duncan Sands for some very stimulating e.mail conversations that clarified a number of difficult questions concerning the dynamics of unimodal maps. The authors are of course solely responsible for any remaining errors or misunderstandings. 2 Running head: Implicit Dynamics Corresponding author: Alfredo Medio, Department of Statistics, University of Udine, Via Treppo 18, 33100, Udine, Italy, Telephone: +39 0432 249584, Fax: +39 0432 249595, e.mail:
[email protected] 3 Abstract We discuss the problem known in economics as \backward dynamics (BD)" occurring in models of perfect foresight, intertemporal equilibrium described mathematically by implicit difference equations. In a previously published paper (Medio and Raines, 2007) we showed that by means of certain mathematical methods and results known as \inverse limits theory (ILT)" it is possible to establish a correspondence between backward dynamics of a non-invertible map and forward dynamics of a related, in- vertible map acting on an appropriately defined space of sequences, each of whose elements corresponds to an intertemporal equilibrium. We also proved the existence of different types of topological attractors for one{dimensional models of overlapping generations. In this paper, we provide an extension of those results, constructing a Lebesgue{like probability measure on spaces of infinite sequences that allows us to distinguish typical from exceptional dynamical behaviors in a measure{theoretical sense, thus proving that all the topological attractors considered in MR07 are also metric attractors.