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We develop a new approach to the computation of the Hausdorff dimension ofthe invariant set of an iterated function system or IFS. In the one dimensionalcase, our methods require only C^3 regularity of the maps in the IFS. The keyidea, which has been known in varying degrees of generality for many years, isto associate to the IFS a parametrized family of positive, linear,Perron-Frobenius operators L_s. The operators L_s can typically be studied inmany different Banach spaces. Here, unlike most of the literature, we study L_sin a Banach space of real-valued, C^k functions, k >= 2; and we note that L_sis not compact, but has a strictly positive eigenfunction v_s with positiveeigenvalue lambda_s equal to the spectral radius of L_s. Under appropriateassumptions on the IFS, the Hausdorff dimension of the invariant set of the IFSis the value s=s_* for which lambda_s =1. This eigenvalue problem is thenapproximated by a collocation method using continuous piecewise linearfunctions (in one dimension) or bilinear functions (in two dimensions). Usingthe theory of positive linear operators and explicit a priori bounds on thederivatives of the strictly positive eigenfunction v_s, we give rigorous upperand lower bounds for the Hausdorff dimension s_*, and these bounds converge tos_* as the mesh size approaches zero.

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