Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms

RESUMO

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 $ \mathrm{dim}(\mu)={h_\mu^{\mathrm{F}}(\nu)}/{(-\lambda)}.$ No discreteness assumption is imposed on the driving measure.