2016/01/25 by Richard S. Falk, Falk, Richard S., Roger D. Nussbaum +1 · 1 citation
Mathematics · Engineering · #Mathematical Analysis and Transform Methods #Algebraic and Geometric Analysis #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.1601.06737
We develop a new approach to the computation of the Hausdorff dimension of\nthe invariant set of an iterated function system or IFS. In the one dimensional\ncase, our methods require only C3 regularity of the maps in the IFS. The key\nidea, which has been known in varying degrees of generality for many years, is\nto associate to the IFS a parametrized family of positive, linear,\nPerron-Frobenius operators Ls. The operators Ls can typically be studied in\nmany different Banach spaces. Here, unlike most of the literature, we study Ls\nin a Banach space of real-valued, Ck functions, k >= 2; and we note that Ls\nis not compact, but has a strictly positive eigenfunction vs with positive\neigenvalue lambdas equal to the spectral radius of Ls. Under appropriate\nassumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS\nis the value s=s_* for which lambdas =1. This eigenvalue problem is then\napproximated by a collocation method using continuous piecewise linear\nfunctions (in one dimension) or bilinear functions (in two dimensions). Using\nthe theory of positive linear operators and explicit a priori bounds on the\nderivatives of the strictly positive eigenfunction vs, we give rigorous upper\nand lower bounds for the Hausdorff dimension s_*, and these bounds converge to\ns_* as the mesh size approaches zero.\n