Cᵐ Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension

الأصل · EN

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 C³ 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 Lₛ. The operators Lₛ can typically be studied in many different Banach spaces. Here, unlike most of the literature, we study Lₛ in a Banach space of real-valued, Cᵏ functions, k >= 2; and we note that Lₛ is not compact, but has a strictly positive eigenfunction vₛ with positive eigenvalue lambdaₛ equal to the spectral radius of Lₛ. Under appropriate assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS is the value s=s* for which lambdaₛ =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 vₛ, we give rigorous upper and lower bounds for the Hausdorff dimension s*, and these bounds converge to s* as the mesh size approaches zero.