$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $mathbb R^1$
$C^m$ Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in $mathbb R^1$
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$C^m$ Perron–Frobenius 算子的特征函数和 Hausdorff 维数值计算的新方法：在 $mathbb R^1$ 中的应用
DOI：
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发表时间：
2016
影响因子：
0.8
通讯作者：
R. Nussbaum
中科院分区：
文献类型：
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作者：
R. S. Falk;R. Nussbaum
我们开发了一种新的方法来计算的Hausdorff维数不变集的迭代函数系统或IFS。在一维情况下，我们的方法只需要IFS中映射的C^3正则性。其关键思想，这已经知道在不同程度的一般性多年来，是关联到IFS的参数化家庭的积极的，线性的，Perron-Frobenius运营商L_s。算子L_s可以在许多不同的Banach空间中进行研究。与大多数文献不同的是，本文研究了实值C^k函数（k >= 2）的Banach空间中的L_s，并注意到L_s不是紧的，而是有一个严格正的本征函数v_s，其正本征值λ_s等于L_s的谱半径。在对IFS的适当假设下，IFS的不变集的Hausdorff维数是值s=s_*，其中lambda_s =1。这个特征值问题，然后近似配置方法使用连续分段线性函数（一维）或双线性函数（二维）。利用正线性算子理论和严格正本征函数v_s的导数的先验界，给出了Hausdorff维数s_* 的严格上下界，当网格尺寸趋于零时，这些界收敛于s_*.
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require only C^3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linear, Perron-Frobenius operators L_s. The operators L_s can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study L_s in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s is not compact, but has a strictly positive eigenfunction v_s with positive eigenvalue lambda_s equal to the spectral radius of 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 lambda_s =1. This eigenvalue problem is then approximated by a collocation method using continuous piecewise linear functions (in one dimension) or bilinear functions (in two dimensions). Using the theory of positive linear operators 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 to s_* as the mesh size approaches zero.