论文标题
非线性横向非符号薄电磁阀的Hausdorff和填料维
Hausdorff and packing dimensions and measures for nonlinear transversally non-conformal thin solenoids
论文作者
论文摘要
We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $Λ$ over the circle嵌入在$ \ mathbb {r}^3 $非共同吸引在稳定的光盘$ w^s $方向上，以非线性吸引到非线性。在假设适当吉布斯的lyapunov指数上的假设和假设下，假设还有一个不变的$ c^{1+c^{1+c^{1+ \ varepsilon} $ strong stable foliation，我们会提出所有量子$ { $ w^s $及其他$ {\ rm hd}（λ）= t_0+1 $。我们还证明，对于包装度量，$ 0 <π_{t_0}（λ\ cap w^s）<\ infty $，但对于Hausdorff Measuion $ {\ rm HM} _ {t_0}（T_0}（λ\ cap w^s）= 0 $ w^s $。 $ 0<π_{1+t_0}（λ）<\ infty $和$ {\ rm hm} _ {1+t_0}（λ）= 0 $。一个技术部分说，不稳定的叶面的单位是本地的Lipschitz，除了一组不稳定的叶子，它们与每$ W^s $的相交都具有量度$ {\ rm hm} _ {t_0} $等于0，甚至hausdorff dounsions $ bessdorff d_0 $。由于现象较大，后者成立。
We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$ hyperbolic, (partially) linear solenoids $Λ$ over the circle embedded in $\mathbb{R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear ones. Under an assumption of transversality and assumptions on Lyapunov exponents for an appropriate Gibbs measure imposing \emph{thinness}, assuming also there is an invariant $C^{1+\varepsilon}$ strong stable foliation, we prove that Hausdorff dimension ${\rm HD}(Λ\cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else ${\rm HD}(Λ)=t_0+1$. We prove also that for the packing measure $0<Π_{t_0}(Λ\cap W^s)<\infty$ but for Hausdorff measure ${\rm HM}_{t_0}(Λ\cap W^s)=0$ for all $W^s$. Also $0<Π_{1+t_0}(Λ) <\infty$ and ${\rm HM}_{1+t_0}(Λ)=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure ${\rm HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.
