ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 39, 50 Goodman, Gerald S., University of Florence, Italy Dilation equations and Hutchinson fractals One way to solve the dilation equation φ(x) = ∑_{k=0}^{m} a_{k}φ(2x − k) is to construct from the mask a_{0}, ..., a_{m} two m × m dilation matrices, P_{0} and P_{1}, whose i, j-th elements are a_{2j−i−1} and a_{2j−1}, resp., where a_{k} = 0 for k < 0 or k > m. Under suitable conditions on the mask, the backwards products P_{d_{n}} … P_{d_{1}} converge for any sequence d_{1}, d_{2}, ... of zeroes and ones to a continuous function f mapping [0, 1] onto the flat Π_{m} in R^{m} made up of row-vectors whose components sum to one. The coordinates of f then determine φ. We prove that from certain masks, including Daubechies’ D_{4}, we can construct a translation-invariant metric on Π_{m} in such a way that each P is contractive with Lipschitz constant equal to its largest subdominant eigenvalue. This makes the graph of f into a Hutchinson fractal and has implications for the rate of convergence of the matrix products and for the Hoelderianity of φ.