标题： 局部逆测度论熵对于自映射 显示英文标题

标题： Local inverse measure-theoretic entropy for endomorphisms

摘要： 我们引入了一个新的概念，即在紧致度量空间上由自映射（非可逆映射）保持的遍历测度的后向轨迹上的局部逆度量熵。 通过使用可测分划定义了第二个逆测度熵的概念。 我们的概念有多个有用的应用。 当它们具有相同的前向测度熵时，逆熵可以区分勒贝格空间上自映射的同构类。 在一个一般情况下，我们证明了遍历测度\mu 的局部逆熵等于前向熵减去折叠熵。 探讨了紧流形上双曲测度的逆熵，重点在于它们的负李雅普诺夫指数。 接下来我们计算了双曲排斥子上逆SRB测度的逆熵。 我们证明了特殊Anosov自映射在\mathbb T^2 上的熵刚性结果，即通过知道其SRB测度的熵和其逆SRB测度的逆熵，可以按光滑共轭进行分类。 接下来我们研究了我们的逆测度论熵与预历史子集上的广义拓扑逆熵之间的关系。 一般来说，我们建立了逆熵的部分变分原理。 在特殊TA覆盖映射在环面上的情况下，我们也得到了逆熵的完整变分原理。 最后，研究了几种自映射的例子，如胖巴克变换、胖索动力吸引子、特殊Anosov自映射、环面自映射，并计算了它们的SRB测度的局部逆熵。 显示英文摘要

摘要： We introduce a new notion of local inverse metric entropy along backward trajectories for ergodic measures preserved by endomorphisms (non-invertible maps) on a compact metric space. A second notion of inverse measure entropy is defined by using measurable partitions. Our notions have several useful applications. Inverse entropy can distinguish between isomorphism classes of endomorphisms on Lebesgue spaces, when they have the same forward measure-theoretic entropy. In a general setting we prove that the local inverse entropy of an ergodic measure \mu is equal to the forward entropy minus the folding entropy. The inverse entropy of hyperbolic measures on compact manifolds is explored, focusing on their negative Lyapunov exponents. We compute next the inverse entropy of the inverse SRB measure on a hyperbolic repellor. We prove an entropy rigidity result for special Anosov endomorphisms of \mathbb T^2, namely that they can be classified up to smooth conjugacy by knowing the entropy of their SRB measure and the inverse entropy of their inverse SRB measure. Next we study the relations between our inverse measure-theoretic entropy and the generalized topological inverse entropy on subsets of prehistories. In general we establish a Partial Variational Principle for inverse entropy. We obtain also a Full Variational Principle for inverse entropy in the case of special TA-covering maps on tori. In the end, several examples of endomorphisms are studied, such as fat baker transformations, fat solenoidal attractors, special Anosov endomorphisms, toral endomorphisms, and the local inverse entropy is computed for their SRB measures.