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