论文标题
用于高维空间上螺线管吸引子维度的二分法
A Dichotomy for the dimension of solenoidal attractors on high dimensional space
论文作者
论文摘要
我们研究由偏斜产品生成的动力系统：$$ t：[0,1）\ times \ mathbb {c} \ to [0,1）\ times \ times \ mathbb {c} \ quad \ quad \ quad \ quad \ quad t（x，y）=（bx \ mod1，bx \ mod1，γy+γ+ϕ（x）） $γ\ in \ mathbb {c} $，这样$ 0 <|γ| <1 $，而$ ϕ $是一个真实的分析$ \ mathbb {z} $ - 定期函数。令[0,1）$中的$δ\，以便$γ= |γ| e^{2πiδ} $。对于情况，$δ\ notin \ mathbb {q} $，我们证明了螺旋螺旋吸引子$ k^ϕ__ {b，\，γ} $ for $ t $：$ k^ϕ__ {b，\，\，\，γ} $是真实的分析函数，或$ k.等于$ \ min \ {3,1+ \ frac {\ log b} {\ log1/|γ|} \} $。此外，鉴于$ b $和$ ϕ $，除非$ ϕ $是恒定的，否则以前的替代方案仅对可数$γ$进行。
We study dynamical systems generated by skew products: $$T: [0,1)\times\mathbb{C}\to [0,1)\times\mathbb{C} \quad\quad T(x,y)=(bx\mod1,γy+ϕ(x))$$ where integer $b\ge2$, $γ\in\mathbb{C}$ such that $0<|γ|<1$, and $ϕ$ is a real analytic $\mathbb{Z}$-periodic function. Let $Δ\in[0,1) $ such that $γ=|γ|e^{2πiΔ}$. For the case $Δ\notin\mathbb{Q}$ we prove the following dichotomy for the solenoidal attractor $K^ϕ_{b,\,γ}$ for $T$: Either $K^ϕ_{b,\,γ}$ is a graph of real analytic function, or the Hausdorff dimension of $K^ϕ_{b,\,γ}$ is equal to $\min\{3,1+\frac{\log b}{\log1/|γ|}\}$. Furthermore, given $b$ and $ϕ$, the former alternative only happens for countable many $γ$ unless $ϕ$ is constant.
