By K.M. Koh, H.P. Yap
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Additional info for Graph Theory Singapore 1983
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, ! D?
KC1 2 / 2 . /C3 . 3) . n Corollary. l/ (here we have not only congruence but in fact equality, since in this case, 21 D 1). Proof of the theorem. 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 !
A/ belongs to S. b/ iff one sequence can be obtained from the other by substituting the tail of the form xyyyy : : : by the tail yxxxx : : : . 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 .
Graph Theory Singapore 1983 by K.M. Koh, H.P. Yap