Uniform measures on inverse limit spaces

Abstract

Abstract. Motivated by problems from dynamic economic models, we consider the problem of defining a uniform measure on inverse limit spaces. Let f : X → X where X is a compact metric space and f is continuous, onto and piecewise one-to-one and Y := lim ←− (X, f ). Then starting with a measure µ 1 on the Borel sets B(X), we recursively construct a sequence of probabil-and n ∈ N. This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y . If µ 1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recursively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evaluation and in dynamical systems in providing an ambient measure (when Lebesgue measure is not available) with which to define an SRB measure or a metric attractor for the shift map on the inverse limit space.

Inverse limits and statistical properties for chaotic implicitly defined economic models

“…We discuss now the notion of utility function on the set of intertemporal equilibria (see for eg. [8], [24]). Definition 4.…”

“…For instance in the cash-in-advance model (see [8], [9], [24], etc) the government is controlling controls the money supply on the market by the growth rule M t+1 = (1 + θ)M t , where θ > 0 is the growth rate. For each θ there exists a different invariant interval [x l (θ), x r (θ)] and inverse limit space Λ(θ).…”