The existence of sigma-finite invariant measures, applications to real one-dimensional dynamics

Abstract

A general construction for $\sigma -$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f_{\ast }^n(\lambda )$ will imply the existence of a $\sigma -$finite invariant measure for the map $f$ which is absolutely continuous with respect to $\lambda$, a measure on the phase space describing the sets of measure zero. Furthermore we will discuss sufficient conditions for the existence of $\sigma -$finite invariant absolutely continuous measures for real 1-dimensional dynamical systems.