C Eigenfunctions of Perron-frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension
نویسندگان
چکیده
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 λs 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 λs = 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.
منابع مشابه
C Eigenfunctions of Perron-frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Applications in R
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 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, linea...
متن کاملC Eigenfunctions of Perron-frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Complex Continued Fractions
In a previous paper [11], 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 ...
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