Title:         C^m Eigenfunctions of Perron-Frobenius Operators and a 
               New Approach to Numerical Computation of Hausdorff Dimension

Authors:       Richard S. Falk and Roger D. Nussbaum

Source:        http://arxiv.org/abs/1601.06737

Status:        

Abstract:      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.

Key words:  Hausdorff dimension, positive transfer operators, 
            continued fractions   

AMS(MOS) subject classifications. Primary 11K55, 37C30; Secondary: 65D05