2016/12/02 by Richard S. Falk, Falk, Richard S., Roger D. Nussbaum +1
Mathematics · #11K55 #37C30 (Primary) 65D05 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.1612.00869
openalex publication_date 2016/12/02 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
In a previous paper, dealing with "Applications in \ℝ1," the\nauthors developed a new approach to the computation of the Hausdorff dimension\nof the invariant set of an iterated function system or IFS and studied some\napplications in one dimension. The key idea, which has been known in varying\ndegrees of generality for many years, is to associate to the IFS a parametrized\nfamily of positive, linear, Perron-Frobenius operators Ls. In our context,\nLs is studied in a space of Cm functions and is not compact.\nNevertheless, it is has a strictly positive Cm eigenfunction vs with\npositive eigenvalue \λs equal to the spectral radius of Ls. 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. To compute the\nHausdorff dimension of an IFS associated to complex continued fractions, (which\nmay arise from an infinite iterated function system), we again approximate the\neigenvalue problem by a collocation method, but now using continuous piecewise\nbilinear functions. Using the theory of positive linear operators and explicit\na priori bounds on the partial derivatives of the strictly positive\neigenfunction vs, we are able to give rigorous upper and lower bounds for\nthe Hausdorff dimension s_*, and these bounds converge to s_* as the mesh\nsize approaches zero. We also demonstrate by numerical computations that\nimproved estimates can be obtained by the use of higher order piecewise tensor\nproduct polynomial approximations, although the present theory does not\nguarantee that these are strict upper and lower bounds. An important feature of\nour approach is that it also applies to the much more general problem of\ncomputing approximations to the spectral radius of positive transfer operators,\nwhich arise in many other applications.\n