混沌驱动凹映射的稳定性指标
Stability index for chaotically driven concave maps
摘要：
本文研究了由双曲基映射S（如baker映射或Anosov曲面同态）驱动的斜积系统，在区间[0，a]上具有简单凹纤维映射h（x）=g（θ）tanh（x），其中g（θ）是由基映射驱动的因子.纤维吸引子是一个上连续函数的图形。对于函数g的许多选择，k有一个零的剩余集，但k>0几乎处处w.r. t。S^（-1）的Sinai-Ruelle-Bowen测度。在这种情况下，我们用局部指数Γ（θ）：= \lim_{n\to\infty} 1/n log g_n（θ）和Λ（θ）：= \lim_{n\to\infty} 1/n \log来估计系统的全局吸引子（即系统的子图）在所有正则点（θ，0）处的稳定性指数|D_u S^{-n}（θ）|以及与S^（-1）和g有关的某个热力学压力函数的正零点s_*。(In在排队论中，s_* 的一个类比称为Loyne指数。Podvigina和Ashwin在2011年引入了稳定性指数，以量化吸引力盆地的局部缩放。
英文摘要：
We study skew product systems driven by a hyperbolic base map S (e.g. a baker map or an Anosov surface diffeomorphism) and with simple concave fibre maps on an interval [0,a] like h(x)=g(θ) tanh(x) where g(θ) is a factor driven by the base map. The fibre-wise attractor is the graph of an upper semicontinuous function ϕ(θ). For many choices of the function g, ϕhas a residual set of zeros but ϕ>0 almost everywhere w.r.t. the Sinai-Ruelle-Bowen measure of S^(-1). In such situations we evaluate the stability index of the global attractor of the system, which is the subgraph of ϕ, at all regular points (θ,0) in terms of the local exponents Γ(θ):=\lim_{n\to\infty} 1/n log g_n(θ) and Λ(θ):=\lim_{n\to\infty} 1/n\log|D_u S^{-n}(θ)| and of the positive zero s_* of a certain thermodynamic pressure function associated with S^(-1) and g. (In queuing theory, an analogon of s_* is known as Loyne's exponent.) The stability index was introduced by Podvigina and Ashwin in 2011 to quantify the local scaling of basins of attraction.
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