Invariant measures and uniform positive entropy property for inverse limits
For a continuous surjective map \(f\) on a compact metric space \(X\), a well-known inverse limit \(\overline {f}\) on the space \(\overline {X}=X^{\infty}\) is associated. The following facts are proven in this paper: (1) The invariant Borel probability measures of \((f,X)\) and \((\overline {f},\overline {X})\) are identical up to a homeomorphism; (2) \((f,X)\) has uniformly positive entropy iff so has \((\overline {f},\overline {X})\). Applications are discussed.
scientific article; zbMATH DE number 1367957