2016/12/02 by Richard S. Falk, Falk, Richard S., Roger D. Nussbaum +1
Computer Science · Engineering · Mathematics · #37C30 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Matrix Theory and Algorithms #Number Theory (math.NT) #Numerical methods in inverse problems #Primary 11K55 #Secondary: 65J10
paper · pdf · doi:10.48550/arxiv.1612.00870
openalex publication_date 2016/12/02 · openalex created_date 2022/08/16 · openalex updated_date 2026/07/28
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 that we consider here, our methods require only C3 regularity of the\nmaps in the IFS. The key idea, which has been known in varying degrees of\ngenerality for many years, is to associate to the IFS a parametrized family of\npositive, linear, Perron-Frobenius operators Ls. The operators Ls can\ntypically be studied in many different Banach spaces. Here, unlike most of the\nliterature, we study Ls in a Banach space of real-valued, Ck functions,\nk \≥ 2. We note that Ls is not compact, but has essential spectral radius\n\ρs strictly less than the spectral radius \λs and possesses a\nstrictly positive Ck eigenfunction vs with eigenvalue \λs. Under\nappropriate assumptions on the IFS, the Hausdorff dimension of the invariant\nset of the IFS is the value s=s_* for which \λs =1. This eigenvalue\nproblem is then approximated by a collocation method using continuous piecewise\nlinear functions. Using the theory of positive linear operators and explicit a\npriori bounds on the derivatives of the strictly positive eigenfunction vs,\nwe give rigorous upper and lower bounds for the Hausdorff dimension s_*, and\nthese bounds converge to s_* as the mesh size approaches zero.\n