Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms

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 ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^1(M)$ satisfying a logarithmic $C^1$ moment condition, and let $\mu$ be a $\nu$-stationary ergodic probability measure. If ${\lambda }_{\mathrm{t}\mathrm{o}\mathrm{p}}={\lambda }_{\mathrm{b}\mathrm{o}\mathrm{t}}=\lambda <0,$ then $\mu$ is exact dimensional and $\mathrm{d}\mathrm{i}\mathrm{m}(\mu )=h_{\mu }^{\mathrm{F}}(\nu )\mathrm{/}(-\lambda )\mathrm{.}$ No discreteness assumption is imposed on the driving measure.