A Tale of Two Fractals by A.A. Kirillov

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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 knowing has remained low; such a lot paintings is aimed both at too mainstream an viewers to accomplish any intensity or at too really expert a neighborhood to accomplish frequent use. Written via celebrated mathematician and educator A.A. Kirillov, A story of 2 Fractals is meant to assist bridge this hole, offering an unique therapy of fractals that's instantaneously available to rookies and sufficiently rigorous for critical mathematicians. The paintings is designed to offer younger, non-specialist mathematicians an excellent starting place within the conception of fractals, and, within the technique, to equip them with publicity to numerous geometric, analytical, and algebraic instruments with functions throughout different areas.

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K / D k 1 for k 1 a -string. 0 D We want to extend a function f 2 Vn n in such a way that the extended function nC1 belongs to f 2 VnC1 . From Eq. 6), we know that this is possible only if n and nC1 are in the same -string. ) belongs to Vn n for all n. So, the following problem arises: is such function f on S1 uniformly continuous and hence can be extended by continuity to S? When this is the case, we can consider the extended function fQ as an eigenfunction for the Laplacian on the whole gasket and define the corresponding eigenvalue as the limit of a suitably renormalized sequence f n g.

Consider the system S with the base b D 2 and digits f0; 1g. S / D Œ 23 ; 13 . 2). 2. Introduce a system S with the base b D 1 C i and digits f0; 1g. S / D ZŒi , which consists of numbers of the form a C i b, a; b 2 Z. These are called the Gaussian integers. S /, it is a fractal compact 50 3 Harmonic Functions on the Sierpi´nski Gasket Fig. 1 C F / : Here, as always, when an arithmetic operation is applied to a set, it means that it is applied to each element of the set. A picture of this set is shown in Fig.

Denote by V the space of smooth functions on with compact support restricted by some boundary conditions; see below. 9) P @vk 2 where d n x is the standard measure on Rn and jgrad vj2 D nkD1 j @x j . 11) @ where @ is the normal derivative and d n 1 y is a measure on the boundary @ . Suppose we restrict v by an appropriate boundary condition that forces the boundary integral in Eq. 11) to vanish. Then the operator  will be exactly the ratio of Q1 and Q0 . 13) In both cases,  is a nonnegative self-adjoint operator in L2 .

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