Singular-hyperbolic attractors are chaotic
Singular-hyperbolic attractors are chaotic
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DOI：
10.1090/s0002-9947-08-04595-9
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发表时间：
2005-11
影响因子：
1.3
通讯作者：
V. Araújo;M. Pacifico;E. Pujals;M. Viana
中科院分区：
文献类型：
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作者：
V. Araújo;M. Pacifico;E. Pujals;M. Viana
证明了三维流的奇异双曲吸引子在两种不同的强意义上是混沌的。首先，流是膨胀的：如果两个点始终保持接近，可能通过时间重新参数化，那么它们的轨道重合。其次，在吸引子上存在一个物理测度（或Sinai-Ruelle-Bowen），其遍历盆地覆盖了吸引拓扑盆地的完整勒贝格（体积）测度子集。此外，该测度沿中心不稳定方向具有绝对连续的条件测度，是一个u-Gibbs态，并且是沿强不稳定方向的时间一图的雅可比矩阵的对数的平衡态。这扩展到奇异双曲吸引子的类别，这类吸引子是流的一致双曲（或公理A）吸引子遍历理论的主要元素。特别地，这些结果可以应用于(i)由洛伦兹方程定义的流，（ii）几何洛伦兹流，（iii）在某些共振双同斜环展开中出现的吸引子，（iv）在某些奇异环展开中出现的吸引子，以及(v)在某些奇异双曲但与几何洛伦兹模型具有不同拓扑类型的几何模型中。在所有这些情况下，结果表明这些吸引子是膨胀的，并且具有u-吉布斯态的物理度量。
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is chaotic, in two different strong senses. First, the flow is expansive: if two points remain close at all times, possibly with time reparametrization, then their orbits coincide. Second, there exists a physical (or Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin covers a full Lebesgue (volume) measure subset of the topological basin of attraction. Moreover this measure has absolutely continuous conditional measures along the center-unstable direction, is a u-Gibbs state and is an equilibrium state for the logarithm of the Jacobian of the time one map of the flow along the strong-unstable direction. This extends to the class of singular-hyperbolic attractors the main elements of the ergodic theory of uniformly hyperbolic (or Axiom A) attractors for flows. In particular these results can be applied (i) to the flow defined by the Lorenz equations, (ii) to the geometric Lorenz flows, (iii) to the attractors appearing in the unfolding of certain resonant double homoclinic loops, (iv) in the unfolding of certain singular cycles and (v) in some geometrical models which are singular-hyperbolic but of a different topological type from the geometric Lorenz models. In all these cases the results show that these attractors are expansive and have physical measures which are u-Gibbs states.