The Dimension of a Point: Computability Meets Fractal Geometry
Recent developments in the theory of computing give a canonical way of assigning a dimension to each point of n-dimensional Euclidean space. Computable points have dimension 0, random points have dimension n, and every real number in [0,n] is the dimension of uncountably many points. If X is a reasonably simple subset of n-dimensional Euclidean space (a union of computably closed sets), then the classical Hausdorff dimension of X is just the supremum of the dimensions of the points in X. In this talk I will discuss the meaning of these developments, their implications for both the theory of computing and fractal geometry, and directions for future research.
- 3.Hitchcock, J.M.: Correspondence principles for effective dimensions. Theory of Computing Systems. In: Proceedings of the 29th International Colloquium on Automata, Languages, and Programming, pp. 561–571 (2002) (to appear) (Preliminary version appeared) Google Scholar
- 4.Lutz, J.H.: Dimension in complexity classes. SIAM Journal on Computing 32, 1236–1259 (2003); Preliminary version appeared in Proceedings of the Fifteenth Annual IEEE Conference on Computational Complexity, pp. 158–169 (2000)Google Scholar