Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms

Abstract: We prove exact dimensionality of ergodic stationary measures for random $C<sup>1$ diffeomorphisms in the single negative Lyapunov scale setting. Let $ν$ be a Borel probability measure on $\mathrm{Diff}<sup>1(M)$ satisfying a logarithmic $C<sup>1$ moment condition, and let $μ$ be a $ν$-stationary ergodic probability measure. If $λ<em>{\mathrm{top}} = λ</em>{\mathrm{bot}} = λ&lt;0,$ then $μ$ is exact dimensional and $\mathrm{dim}(μ)={h_μ<sup>{\mathrm{F}}(ν)}/{(-λ)}.$ No discreteness assumption is imposed on the driving measure.