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By Ainouche A., Schiermeyer I.

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I . 6) . /: These relations allow us to develop the arithmetic theory of basic functions for any integer n, not necessarily positive,5 analogously to the case n D 2. t/ always takes integer values at integer points. I do not know of a geometric interpretation of these functions for n Ä 0 as harmonic functions of some kind. 5 48 3 Harmonic Functions on the Sierpi´nski Gasket Some values of n are of special interest. t/ D 2t t 2 . 9. 9) A detailed investigation of this function is very interesting, and I would highly recommend it for an independent study.

8. Which infinite sequences correspond (a) (b) (c) (d) to boundary points? to points of segments joining the boundary points? to vertices of Sn ? 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, !

Consider the triangular piece of the infinite gasket that is based on the segment Œk 1; k C 1. It is shown in Fig. 4. We denote the values of at the points k 1; k; k C1 by a ; a; aC respectively. Then the values bC ; b ; c in the remaining vertices shown in Fig. l/ is an integer when l < 2n . 42 3 Harmonic Functions on the Sierpi´nski Gasket The result is c D 5a 2a 3a C 2aC ; 5 bC D 2a 2aC ; b D 2a 2aC C 3a : 5 Consider now the functions g˙ W ! k ˙ /. Knowing the boundary values of the corresponding harmonic functions on pieces of S, we can write a˙ C b˙ 2 g˙ .

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