arXiv 1701.083422017-01math.DS

具有absolutely continuous不变概率的高维吸引子

Higher-dimensional attractors with absolutely continuous invariant probability

Carlos Bocker-Neto, Ricardo Bortolotti

备注：: 19 pages
DOI：: 10.1088/1361-6544/aaaaf6

摘要：

考虑由$ T(x,y) = (E(x), C(y) + f(x))$给出的动力系统$T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow \mathbb{T}\times \mathbb{R}^{d} $，其中$E$是$\mathbb{T}$的线性扩张映射，$C$是$\mathbb{R}^d$的线性收缩映射，$f$在$C^2(\mathbb{T},\mathbb{R}^d)$中.我们证明了如果$T$是体积膨胀的，$u\geq d$，那么对于每个$E$都存在一个开集$\mathcal{U}$的对$(C,f)$，相应的动态$T$承认一个绝对连续不变的概率。在$\mathcal{U}$中的映射的动态中存在$\mathbb{T}\times\{ 0 \}$的图像的自相交之间的横截性的几何特征。另外，我们给出了一个介于$E$和$C$之间的条件，在这个条件下，扰动$f$在$\mathcal{U}$中得到一对$(C,\tilde{f})$是可能的.

英文摘要：

Consider a dynamical system $T:\mathbb{T}\times \mathbb{R}^{d} \rightarrow \mathbb{T}\times \mathbb{R}^{d} $ given by $ T(x,y) = (E(x), C(y) + f(x))$, where $E$ is a linear expanding map of $\mathbb{T}$, $C$ is a linear contracting map of $\mathbb{R}^d$ and $f$ is in $C^2(\mathbb{T},\mathbb{R}^d)$. We prove that if $T$ is volume expanding and $u\geq d$, then for every $E$ there exists an open set $\mathcal{U}$ of pairs $(C,f)$ for which the corresponding dynamic $T$ admits an absolutely continuous invariant probability. A geometrical characteristic of transversality between self-intersections of images of $\mathbb{T}\times\{ 0 \}$ is present in the dynamic of the maps in $\mathcal{U}$. In addition, we give a condition between $E$ and $C$ under which it is possible to perturb $f$ to obtain a pair $(C,\tilde{f})$ in $\mathcal{U}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文