By A.A. Kirillov

Since Benoit Mandelbrot's pioneering paintings within the past due Nineteen Seventies, ratings of study articles and books were released relating to fractals. regardless of the quantity of literature within the box, the overall point of theoretical realizing has remained low; so much paintings is aimed both at too mainstream an viewers to accomplish any intensity or at too really expert a group to accomplish frequent use. Written by way of celebrated mathematician and educator A.A. Kirillov, *A story of 2 Fractals* is meant to aid bridge this hole, supplying an unique remedy of fractals that's immediately obtainable to novices and sufficiently rigorous for critical mathematicians. The paintings is designed to provide younger, non-specialist mathematicians a high-quality starting place within the conception of fractals, and, within the method, to equip them with publicity to various geometric, analytical, and algebraic instruments with purposes throughout different areas.

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**Example text**

8. The derivative y 0 D function of x. dy dx exists and is a continuous strictly decreasing We leave the proof to the reader as a rather nontrivial exercise. 4. t// 0 Hint. 14). x/ explicitly at some points (knowing that the derivative exists). 0/C1 0 2 0 1 D 0. 0/ This quadratic equation has two roots: 13 and 1. But since y. 1/ D 0 and y. 1 C "/ > 0, only the first root is suitable. So we get y 0 . 1/ D 13 . 1/ D 13 . x/ are shown in Fig. 5. t/ with a rational t. Indeed, every rational number r can be written as an eventually periodic dyadic fraction.

The next problem is open. 4. x/. It contains a big subset X of points with rational coordinates. For instance, all the points that correspond to the rational values of the parameter t belong to X . It is of great interest to study the closure X p in the p-adic topology (see Info G below). 7 The Harmonic Image of S To conclude the first part of the book, we show how the Sierpi´nski gasket is related to the Apollonian gasket—the main subject of the second part. p Let us introduce a complex harmonic function z D f i1 ; 13 on S.

FS92]). For illustration and to make things clear, we consider in detail the case n D 2. D/ First let V D V2 . It is a 3-dimensional space of functions on S2 whose values are shown in Fig. 1. 4x y z; 4y Áx z; 4z x y/. In the natural basis, this operator is given by the matrix 4 1 1 1 4 1 . The eigenvalues can be easily computed using the following lemma. 1. 1 The Laplace Operator on Sn 25 Fig. 1 Functions on S2 with Dirichlet condition 0 y x z 0 Fig. n 1/b. 0 b with multiplicity n y+z 2 1 and one more eigenvalue In our case, we have a double eigenvalue 5 and simple eigenvalue 2.