非均匀共形收缩随机微分同胚的平稳测度的精确维数
Exact dimensionality of stationary measures for nonuniformly conformally contracting random diffeomorphisms
- University of Chicago（芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明在单一负李雅普诺夫指数条件下，非均匀共形收缩随机微分同胚的平稳测度具有精确维数，其维数由Furstenberg熵与负指数之比给出，且无需离散性假设。
中文摘要 AI 辅助
我们证明了在单一负李雅普诺夫指数尺度设置下，随机$C^1$微分同胚的遍历平稳测度具有精确维数。设$\nu$为$\mathrm{Diff}^1(M)$上满足对数$C^1$矩条件的Borel概率测度，且$\mu$为$\nu$-平稳遍历概率测度。若$\lambda_{\mathrm{top}} = \lambda_{\mathrm{bot}} = \lambda<0,$ 则$\mu$是精确维数的，且$ \mathrm{dim}(\mu)={h_\mu^{\mathrm{F}}(\nu)}/{(-\lambda)}.$ 对驱动测度不施加离散性假设。
英文摘要
We prove exact dimensionality of ergodic stationary measures for random $C^1$ diffeomorphisms in the single negative Lyapunov scale setting. Let $ν$ be a Borel probability measure on $\mathrm{Diff}^1(M)$ satisfying a logarithmic $C^1$ moment condition, and let $μ$ be a $ν$-stationary ergodic probability measure. If $λ_{\mathrm{top}} = λ_{\mathrm{bot}} = λ<0,$ then $μ$ is exact dimensional and $ \mathrm{dim}(μ)={h_μ^{\mathrm{F}}(ν)}/{(-λ)}.$ No discreteness assumption is imposed on the driving measure.
