aa r X i v : . [ m a t h . C A ] S e p Quasi-fractal sets in space
Stephen SemmesRice University
Let a be a positive real number, a < /
2. A standard construction of aself-similar Cantor set in the plane starts with the unit square [0 , × [0 , a , then replaces eachof those squares with their four corner squares of sidelength a , and so on.At the n th stage one has 4 n squares with sidelength a − n , and the resultingCantor set has Hausdorff dimension log 4 / ( − log a ). The limiting case a =1 / a < /
4. The Cantor set may bedescribed as the singular part of this quasi-fractal set, which is compact andconnected.Of course, one can consider similar constructions in higher dimensions.For the sake of simplicity, let us focus on connected fractal sets in R withtopological diimension 1, for which the corresponding quasi-fractal set isobtained by including a countable collection of 2-dimensional pieces. As inthe case of higher-dimensional Sierpinski gaskets or Menger sponges, theseadditional 2-dimensional pieces could be triangles or squares. Just as theendpoints of the line segments were elements of the Cantor set in the previoussituation, the boundaries of these two-dimensional pieces would be loops inthe fractal.For that matter, one could do the same for connected fractals in theplane like Sierpinski gaskets and carpets, where the additional 2-dimensionalpieces are simply the bounded components of the complement. The resultingquasi-fractal would then be an ordinary 2-dimensional set, such as a triangleor a square. In each of these situations, the fractal set is the topologicalboundary of the rest, which is disconnected but smooth. One could alsolook at the bounded components of the complement of the quasi-fractals in R , but we shall not pursue this here.1his type of connected fractal set in the plane or space has a lot of1-dimensional topological activity, which can be described in terms of ho-motopy classes of continuous mappings to the circle. In particular, thereare indices for such mappings associated to loops in the fractal. This sug-gests looking at Toeplitz operators and their indices when they are Fred-holm, using Bergman or Hardy spaces of holomorphic functions on the two-dimensional smooth part.One already has complex analysis sitting on the plane, and for the 2-dimensional smooth pieces in space there are conformal structures inducedby the ambient Euclidean geometry. Once orientations are chosen, one getscomplex structures, and one can talk about Bergman and Hardy spaces.Different choices of orientations lead to different types of Toeplitz oper-ators. It is natural to have a lot of different types of Toeplitz operators inthis situation, because of the complexity of the fractals. References [1] W. Arveson,
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