## Abstract

In a previous paper (Falk and Nussbaum, in C^{m} Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of hausdorff dimension: applications in R^{1}, 2016. ArXiv e-prints arXiv:1612.00870), the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS and studied some applications in one dimension. 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}. In our context, L_{s} is studied in a space of C^{m} functions and is not compact. Nevertheless, it has a strictly positive C^{m} eigenfunction v_{s} with positive eigenvalue λ_{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 λ_{s}= 1. To compute the Hausdorff dimension of an invariant set for an IFS associated to complex continued fractions, (which may arise from an infinite iterated function system), we approximate the eigenvalue problem by a collocation method using continuous piecewise bilinear functions. Using the theory of positive linear operators and explicit a priori bounds on the partial derivatives of the strictly positive eigenfunction v_{s}, we are able to give rigorous upper and lower bounds for the Hausdorff dimension s_{∗}, and these bounds converge to s_{∗} as the mesh size approaches zero. We also demonstrate by numerical computations that improved estimates can be obtained by the use of higher order piecewise tensor product polynomial approximations, although the present theory does not guarantee that these are strict upper and lower bounds. An important feature of our approach is that it also applies to the much more general problem of computing approximations to the spectral radius of positive transfer operators, which arise in many other applications.

Original language | English (US) |
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Article number | 61 |

Journal | Integral Equations and Operator Theory |

Volume | 90 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2018 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory

## Keywords

- Continued fractions
- Hausdorff dimension
- Positive transfer operators