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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 that we consider here, our methods require only $C^3$ regularity of themaps in the IFS. The key idea, which has been known in varying degrees ofgenerality for many years, is to associate to the IFS a parametrized family ofpositive, linear, Perron-Frobenius operators $L_s$. The operators $L_s$ cantypically be studied in many different Banach spaces. Here, unlike most of theliterature, 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 astrictly positive $C^k$ eigenfunction $v_s$ with eigenvalue $\lambda_s$. Underappropriate assumptions on the IFS, the Hausdorff dimension of the invariantset of the IFS is the value $s=s_*$ for which $\lambda_s =1$. This eigenvalueproblem is then approximated by a collocation method using continuous piecewiselinear functions. Using the theory of positive linear operators and explicit apriori bounds on the derivatives of the strictly positive eigenfunction $v_s$,we give rigorous upper and lower bounds for the Hausdorff dimension $s_*$, andthese bounds converge to $s_*$ as the mesh size approaches zero.

$C^m$ Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Applications in $\mathbb{R}^1$