## Mathematics of FractalsThis book aims at providing a handy explanation of the notions behind the self-similar sets called 'fractals' and 'chaotic dynamical systems'. The authors emphasize the beautiful relationship between fractal functions (such as Weierstrass') and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems. These functions are shown to be in a sense unique solutions of certain boundary problems. The last chapter of the book treats harmonic functions on fractal sets. |

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### Contents

SelfSimilar Sets | 17 |

An Alternative Computation for Differentiation | 33 |

In Quest of Fractal Analysis | 53 |

Recommended Reading | 75 |

### Common terms and phrases

analysis arbitrary ball of radius basic relation box-counting dimension C C(K Cantor set chaotic dynamical system Chapter closed ball coefficient compact sets completes the proof construct continuous function corresponding countable definition denote derivative difference equations dimension function dimT(X Dirichlet problem discrete dynamical system Em(u equality example Figure finite sequences fixed point following theorem formula Fourier function f Geometry graph h(Pl harmonic function Hausdorff dimension Hausdorff measure Hence Hs(X implies induction inequality integer interval K(Rn Koch curve l(Vm Laplacian Lebesgue's singular function Lemma linear Ma(x Masayoshi Hata Math monotone decreasing natural number open set condition Poisson's equation problem for Poisson's proof of Theorem properties real number Rham satisfy the open Schauder expansion self-affine set self-similar set Sierpinski gasket similar contractions singular function solution subinterval subset system of difference Takagi function Theorem 4.1 theory topological dimension Vm+i Vm+i\Vm wavelet Weierstrass Yamaguti zero