管状维数：叶向渐近局部乘积结构，以及熵与体积增长
Tubular dimension: Leaf-Wise Asymptotic Local Product Structure, and Entropy and Volume Growth
AI总结：
本文引入管状维数并给出计算公式，利用该概念证明了微分同胚条件测度的渐近局部乘积结构，给出了条件熵间隙的体积增长界，并计算了特定映射的最大熵测度条件熵，为解决Binder-DeMarco猜想奠定基础。
摘要：
我们引入了管状维数的概念，并给出了其计算公式。作为应用，我们证明了闭Riemann流形上$C^{1+γ}$微分同胚的每个不变测度，在不稳定叶的中间叶状结构上的条件测度具有渐近局部乘积结构。作为第二个应用，我们以体积增长的形式证明了对任意两个相邻条件熵之间间隙的界。作为第三个应用，对于某些$C^\infty$映射，我们计算了最大熵测度的所有条件熵；特别地，作为推论，在后续论文中我们计算了$\mathbb{C}\mathbb{P}^k$（$k\geq 1$）的全纯自同态平衡测度的Hausdorff维数，给出了Binder-DeMarco猜想的解，并回答了Fornæss和Sibony的问题。
英文摘要：
We introduce the notion of tubular dimension, and give a formula for it. As an application we show that every invariant measure of a $C^{1+γ}$ diffeomorphism of a closed Riemannian manifold admits an asymptotic local product structure for conditional measures on intermediate foliations of unstable leaves. As a second application, we prove a bound on the gap between any two consecutive conditional entropies, in the form of volume growth. As a third application, for certain $C^\infty$ maps we compute all conditional entropies for the measure of maximal entropy; And in particular as a consequence, in a follow-up paper we compute the Hausdorff dimension of the equilibrium measure of holomorphic endomorphisms of $\mathbb{C}\mathbb{P}^k$, $k\geq 1$, giving a solution to the Binder-DeMarco conjecture, and answering a question of Fornæss and Sibony.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
