Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms

Abstract

We prove exact dimensionality of ergodic stationary measures for random $C^1$ diffeomorphisms in the single negative Lyapunov scale setting. Let $\nu$ be a Borel probability measure on ${\mathrm{Diff}}^1(M)$ satisfying a logarithmic $C^1$ moment condition, and let $\mu$ be a $\nu$-stationary ergodic probability measure. If ${\lambda }_{\mathrm{top}}={\lambda }_{\mathrm{bot}}=\lambda <0,$ then $\mu$ is exact dimensional and $\dim (\mu )=h_{\mu }^{\mathrm{F}}(\nu )/(-\lambda ).$ No discreteness assumption is imposed on the driving measure.

16 pages