| This thesis studies the optimal parameterizations of self-similar sets, that is, parameterizations that are almost one-to-one, measure-preserving and l/s-Holder continuous, where s is the Hausdorff dimension of the set in concern. In particular, we unveil the mystery of space-filling curves, a prominent object in mathematics, by carrying out a systematic and rigorous treatment. Through this method, we design several exciting programs such that whenever you input data, you can get an optimal parametrization immediately.First, we introduce a notion of linear GIFS, a graph-directed iterated function system equipped with an order structure, and show that the invariant sets of a linear GIFS admit optimal parameterizations if the open set condition is fulfilled. Indeed, constructing of space-filling curves is the same as constructing certain linear GIFS.Second, to exploring the linear GIFS structures of self-similar sets, we introduce the finite skeleton property for an iterated function system (IFS). A finite skeleton determines a complete directed graph Go-Inputting Go and’iterating’it by the IFS, we obtain a graph consisting of affine copies of Go-By studying the relation between Go and the new graph, we can define various edge-substitution rules, which induce various linear GIFS’. The problem of checking the open set condition is handled by the theory of fractal geometry. Hence we prove that with some additional conditions fulfilled, a self-similar set adopts an optimal parametrization.And, the classical space-filling curves are described in more detail and we also give three method to introduce the classical space-filling curves.Finally, we show that the’additional conditions’are superfluous, and hence the open set condition together with the finite skeleton property guarantee the ex-istence of optimal parameterizations. Our results cover a large class of self-similar sets, including self-similar sets of finite type, and extend almost all the previous studies in this direction. |
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