论文标题
SRB度量尺寸的二分法
A Dichotomy for the dimension of SRB measure
论文作者
论文摘要
我们研究由偏斜产品生成的动力系统：$$ t：[0,1）\ times \ mathbb {r} \ to [0,1）\ times \ times \ mathbb {r} \ quad \ quad \ quad \ quad \ quad \ quad t（x，y）=（bx \ mod1，bx \ mod1，γy+γ+ϕ（x）$ b $ b $ b。实际分析$ \ Mathbb {z} $ - 定期函数。我们证明了$ t $的SRB测量$ω$的以下二分法：$ω$的支持是真实分析功能的图表，或者$ω$的尺寸等于$ \ min \ min \ {2,1+ \ frac \ frac {\ log log b} {\ log1/γ} \} $。此外，给定$ b $和$ ϕ $，除非$ ϕ $是恒定的，否则以前的替代方案仅适用于有限的$γ$。
We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{R}\to [0,1)\times\mathbb{R} \quad\quad T(x,y)=(bx\mod1,γy+ϕ(x))$$ where integer $b\ge2$, $0<γ<1$ and $ϕ$ is a real analytic $\mathbb{Z}$-periodic function. We prove the following dichotomy for the SRB measure $ω$ for $T$: Either the support of $ω$ is a graph of real analytic function, or the dimension of $ω$ is equal to $\min\{2,1+\frac{\log b}{\log1/γ}\}$. Furthermore, given $b$ and $ϕ$, the former alternative only happens for finitely many $γ$ unless $ϕ$ is constant.
