Uniform Measures On Inverse Limit Spaces

Abstract. Motivated by problems from dynamic economic mod-els, 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 µ1 on the Borel sets B(X), we recursively construct a sequence of probabil-ity measures {µn}∞n=1 on B(X) satisfying µn(A) = µn+1[f−1(A)] for each A ∈ B(X) and n ∈ N. This sequence of probability measures is then uniquely extended to a probability measure on the inverse limit space Y. If µ1 is a uniform measure, we argue that the measure induced on the inverse limit space by the recur-sively constructed sequence of measures is a uniform measure. As such, the measure has uses in economic theory for policy evalua-tion 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. 1.