论文标题
经典洛伦兹吸引子和较高尺寸的分节型吸引子的平衡状态
Equilibrium states for the classical Lorenz attractor and sectional-hyperbolic attractors in higher dimensions
论文作者
论文摘要
长期以来，人们一直认为，经典的洛伦兹吸引子支持一种独特的最大熵度量。在本文中，我们通过考虑了Hölder连续功能的平衡状态在截面 - 毛细血管吸引子$λ$上的均衡状态的唯一性，从而为这一猜想及其较高维度提供了积极的答案。我们证明，在$ c^1 $开放和密集的矢量场（包括经典的洛伦兹吸引子）中，如果奇异点的点质量不是平衡状态，那么就存在$λ$的独特平衡状态。特别是，存在$ x |_λ$的最大熵的独特量度。
It has long been conjectured that the classical Lorenz attractor supports a unique measure of maximal entropy. In this article, we give a positive answer to this conjecture and its higher-dimensional counterpart by considering the uniqueness of equilibrium states for Hölder continuous functions on a sectional-hyperbolic attractor $Λ$. We prove that in a $C^1$-open and dense family of vector fields (including the classical Lorenz attractor), if the point masses at singularities are not equilibrium states, then there exists a unique equilibrium state supported on $Λ$. In particular, there exists a unique measure of maximal entropy for the flow $X|_Λ$.
