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.
翻译:在 [14] 中,作者开发了一种新的方法来计算迭代函数系统或 IFS 的变异功能集的Hausdorf 维度值。 在本文中,我们扩展了这一方法以纳入高顺序近似方法。 我们再次依赖这样的事实:我们可以将一个正、线性、 Perron-Frobenius 操作员的百折叠式的组合与 IFS 挂钩, 这个想法多年来以不同程度的笼统性为名。 尽管在我们所考虑的设置中, 美元不是紧凑的, 但是它拥有一个绝对正正的 美元 egenforf 维度值 $+m 美元 折叠值 美元 和 egeneral liveral $slational $slational $slational legal $slorg $R_ slational $slational legyalal $ral $s a pal legal leas a prial sual subal subal sultal sultal sultal as a pas a pal rocial sultal sual subal subal sual subal subal subal subal subal subal subal subal subal subal subal subal subal subal subal suble suble 美元, 美元, subal a a a a a a a a a a pal subal subal subal subal subal ro subal subal ro ro ro ro ro ro ro ro ro ro roal roal roal roal roal subal roal roal ro) ro) ro, ro, ro ro, ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro ro