论文标题
隐藏的积极性和一种新方法来计算Hausdorff维度：高阶方法
Hidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods
论文作者
论文摘要
在[14]中，作者开发了一种新的方法来计算迭代功能系统不变集或IFS的hausdorff维度。在本文中，我们扩展了这种方法以结合高阶近似方法。我们再次依靠这样一个事实，即我们可以与一个正面的，线性的，perron-frobenius操作员$ l_s $相关联，这是多年来以不同程度的一般性而闻名的想法。尽管在我们考虑的设置中，$ l_s $并不紧凑，但它具有严格的正$ c^m $ eigenfunction $ v_s $ v_s $，with eigenvalue $ r（l_s）$ r（l_s）$ for nutary $ m $，所有其他点$ z $在$ l_s $ l_s $ l_s $ sapplate $ | z | \ le B $，用于某些常数$ b <r（l_s）$。在IFS上的适当假设下，IFS的不变集的Hausdorff尺寸为$ r（l_s）= 1 $的值$ s = s _*$。然后，使用任意度$ r $的连续分段多项式$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $的连续分段多项式，则通过搭配方法在每个次室间点的扩展chebyshev点上近似此特征值问题。利用将perron的正矩阵理论延伸到矩阵上，将锥体$ k $映射到其内部，并明确对严格积极的本征函数的衍生物$ v_s $的先验范围，我们给出了严格的上和下限，为hausdorff dimension $ s _*$和这些bumpers和$ s y y $ s $ s y contime y y y y y y $ s $ sials and $ s $ s y*$ s _*增加。
In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend this approach to incorporate high order approximation methods. We again rely on the fact that we can associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators $L_s$, an idea known in varying degrees of generality for many years. Although $L_s$ is not compact in the setting we consider, it possesses a strictly positive $C^m$ eigenfunction $v_s$ with eigenvalue $R(L_s)$ for arbitrary $m$ and all other points $z$ in the spectrum of $L_s$ satisfy $|z| \le b$ for some constant $b < R(L_s)$. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value $s=s_*$ for which $R(L_s) =1$. This eigenvalue problem is then approximated by a collocation method at the extended Chebyshev points of each subinterval using continuous piecewise polynomials of arbitrary degree $r$. Using an extension of the Perron theory of positive matrices to matrices that map a cone $K$ to its interior and explicit a priori bounds on the derivatives of the strictly positive eigenfunction $v_s$, we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, and these bounds converge rapidly to $s_*$ as the mesh size decreases and/or the polynomial degree increases.
