Limit sets of typical continuous functions

Abstract

Given a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that for the typical continuous function f : X → X , it is true that for every point x in a full μ-measure subset of X the limit set ω ( f , x ) is a Cantor set of Hausdorff dimension zero, f maps ω ( f , x ) homeomorphically onto itself, each point of ω ( f , x ) has a dense orbit in ω ( f , x ) and f is non-sensitive at each point of ω ( f , x ) ; moreover, the function x → ω ( f , x ) is continuous μ-almost everywhere.