Title: C^m Eigenfunctions of Perron-Frobenius Operators and a
New Approach to Numerical Computation of Hausdorff Dimension:
Applications in $\R^1$
Authors: Richard S. Falk and Roger D. Nussbaum
Source: J. Fractal Geometry, vol. 5 (2018), no. 3, 279-337.
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 that we consider here, 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, $k \ge 2$. We note that $L_s$ is not
compact, but has essential spectral radius $\rho_s$ strictly less than the
spectral radius $\lambda_s$ and possesses a strictly positive $C^k$
eigenfunction $v_s$ with eigenvalue $\lambda_s$. 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. 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: 65J10