Noise-induced order from positive Lyapunov exponents

Let $I$ be the phase space and let $T:I\to I$ be a $C^1$, nonsingular dynamical system admitting a unique absolutely continuous invariant measure with positive Lyapunov exponent. Let $m$ denote this measure. Noise-induced order conjecture. If

$$-\infty<\int \log(|T'|)\,dm<0,$$

then the system presents Noise Induced Order, meaning that there exist $0<\xi_1<\xi_2$ such that for every $\xi>\xi_1$ the system has a unique stationary measure with density $f_{\xi}$ and the Lyapunov exponent of the stationary measure satisfies $\lambda(\xi_1)>0$ and $\lambda(\xi_2)<0$. The conjecture proposes a general mechanism linking stochastic stability and transitions in Lyapunov exponents for one-dimensional systems with additive noise; the paper presents the criterion as a conjectural extension of known stochastic-stability results, and its resolution is not supplied here.