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On the computability of fractal dimensions and Hausdorff measure. (English) Zbl 0926.03049
Summary: It is shown that there exist subsets \(A\) and \(B\) of the real line which are recursively constructible such that \(A\) has a nonrecursive Hausdorff dimension and \(B\) has a recursive Hausdorff dimension (between 0 and 1) but has a finite, nonrecursive Hausdorff measure. It is also shown that there exists a polynomial-time computable curve on the two-dimensional plane that has a nonrecursive Hausdorff dimension between 1 and 2. Computability of Julia sets of computable functions on the real line is investigated. It is shown that there exists a polynomial-time computable function \(f\) on the real line whose Julia set is not recursively approximable.

MSC:
03D80 Applications of computability and recursion theory
28A80 Fractals
03D15 Complexity of computation (including implicit computational complexity)
03F60 Constructive and recursive analysis
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
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