GEOMETRIC PROPERTIES OF IDEMPOTENT PROBABILITY MEASURE SPACES: CATEGORICAL, METRIC AND DYNAMICAL ASPECTS
Keywords:
idempotent measure, Maslov measure, Bellman operator, contraction, monad, Hausdorff metric, local convexity, cross-sections, fixed point.Abstract
This paper presents an extended investigation I(X) of the geometric and categorical properties of the space of idempotent probability measures. Building upon earlier results, we prove the following new theorems: 1) the monad structure of the functor I(X) and its relationship with the hyperspace monad; 2) completeness degree and Lipschitz continuity of I(X) under the Hausdorff metric; 3) the strict contraction property of the idempotent Bellman operator and existence of its unique fixed idempotent measure; 4) local convex structure and cross-sectional properties of I(X). The main new result states: if X is a compact metric space and T:C(X)--C(X) is an idempotent Bellman operator, then there exists a unique fixed idempotent measure in I(X) for T.
