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A Tale of Two Fractals by A.A. Kirillov

By A.A. Kirillov

Since Benoit Mandelbrot's pioneering paintings within the past due Nineteen Seventies, rankings of study articles and books were released concerning fractals. regardless of the quantity of literature within the box, the final 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 good 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 assist bridge this hole, delivering an unique therapy of fractals that's instantly available to novices and sufficiently rigorous for severe mathematicians. The paintings is designed to offer younger, non-specialist mathematicians an effective origin within the concept of fractals, and, within the technique, to equip them with publicity to quite a few geometric, analytical, and algebraic instruments with purposes throughout different areas.

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T/ D Bt1 ! z/; ! s/ D Bt1 ! 0; 1. Since the Bi have nonnegative coefficients, it is ! z/. t/ 0:8 ! z/ 0:8 ! 2 Fig. 2. For all x 2 Œ0; 1, we have the relations A 1x˛ Ä with A D Proof. x/ Ä 5 for 1 2nC1 ÄxÄ 1 : 2n But for the given value of ˛, we also have  ÃnC1  Ãn 3 3 Ä x˛ Ä 5 5 for 1 2nC1 ÄxÄ 1 : 2n This implies the first statement of the theorem. The second can be proved in the same way. 1/ D 0 (see Fig. 2). 2 The Functions ; '; ; 35 On the other hand, the functions ; '; ; , being strictly monotone, have a finite derivative at almost all points of the interval Œ0; 1.

So here I restrict myself to a short description of the rather interesting technique used in the study of the spectrum. To define the analogue of a Laplace operator on the Sierpi´nski gasket S, we consider first the finite approximation Sn of S. Let us try to follow the algebraic scheme used above. Let Sn be the nth finite approximation to the Sierpi´nski gasket S. Denote by Vn the set of real functions nC1 on Sn . Since Sn consists of 3 2 C3 points, Vn is a real vector space of dimension dn D 3nC1 C3 .

To segments joining the vertices of Sn ? 0; 1/ in a continuous, order-preserving manner. Later, in 1938, this function was introduced by A. Denjoy for arbitrary real numbers. By definition,7 the function ?. / sends a number a represented by the continued fraction 1 aD 1 a1 C 1 a2 C 1 :: :C 1 ak C :: : to the number X . a/ WD 2a1 C Cak a1 1 a2 a3 ‚ …„ ƒ ‚…„ƒ ‚…„ƒ D 0:0 : : : 0 1 : : : 1 0 : : : 0 : : : : k 1 For example, ! D? 3 7 p Á 1 2C 1 1 3C 1 D 0:0110; D 0:1100 D 45 ; Á X 2 2 2 k: ? ee2 C11 D ?

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