Abstract

From the text: The author notes that if \(T\) is an aperiodic measure preserving transformation on a probability space \((\Omega,\mu)\), then there is an open set in \({\mathcal G}=\{f\in L^\infty(\Omega,\mu)\colon f^{-1}\in L^{\infty}(\Omega,\mu)\}\) such that \(f\) is non-uniformly hyperbolic: \(\int_\Omega \log|f(x)| d\mu(x)>0\) but there exists no \(g\in{\mathcal G}, \epsilon>0\) such that \(g(T(x))f(x) g(x)^{-1} \geq (1+\epsilon)\) for almost all \(x\).