AI 中文总结
研究\(C^{1 + \mathrm{Dini}}\)微分同胚，证明其SRB测度刻画，当支配丛有正Lyapunov指数时条件测度绝对连续性与部分Pesin熵公式等价，得到部分双曲大多扩张吸引子的SRB测度，还指出\(C^1\)范畴不成立。
AI 中文摘要
对于\(C^{1 + \mathrm{Dini}}\)微分同胚，我们证明了在其支撑集允许支配分裂的不变测度中，SRB测度的一种类似Ledrappier - Young型的刻画。当支配丛仅有正Lyapunov指数时，相应Pesin不稳定流形上条件测度的绝对连续性等同于部分Pesin熵公式。作为应用，我们在\(C^{1 + \mathrm{Dini}}\)范畴中得到了部分双曲大多扩张吸引子的SRB测度。还给出反例表明这些结论在\(C^1\)范畴中不成立，即使在一致双曲性下。
英文摘要
For $C^{1+\mathrm{Dini}}$ diffeomorphisms, we prove a Ledrappier--Young-type characterization of SRB measures among invariant measures whose supports admit dominated splittings. When the dominating bundle has only positive Lyapunov exponents, absolute continuity of the conditional measures on the corresponding Pesin unstable manifolds is equivalent to the partial Pesin entropy formula. As an application, we obtain SRB measures for partially hyperbolic mostly expanding attractors in the $C^{1+\mathrm{Dini}}$ category. Counterexamples are provided to show that these conclusions fail in the $C^1$ category, even under uniform hyperbolicity.
