A mathematical gift, 3, interplay between topology, by Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

By Kenji Ueno, Koji Shiga, Shigeyuki Morita, Toshikazu Sunada

This e-book brings the sweetness and enjoyable of arithmetic to the study room. It bargains critical arithmetic in a full of life, reader-friendly sort. integrated are routines and plenty of figures illustrating the most innovations. the 1st bankruptcy talks in regards to the thought of manifolds. It comprises dialogue of smoothness, differentiability, and analyticity, the assumption of neighborhood coordinates and coordinate transformation, and a close rationalization of the Whitney imbedding theorem (both in vulnerable and in powerful form). the second one bankruptcy discusses the proposal of the world of a determine at the airplane and the amount of an excellent physique in house. It comprises the evidence of the Bolyai-Gerwien theorem approximately scissors-congruent polynomials and Dehn's resolution of the 3rd Hilbert challenge. this is often the 3rd quantity originating from a chain of lectures given at Kyoto college (Japan). it's appropriate for school room use for prime institution arithmetic lecturers and for undergraduate arithmetic classes within the sciences and liberal arts.

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Additional info for A mathematical gift, 3, interplay between topology, functions, geometry, and algebra

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The preceding arguments demonstrate that p is not b or e and so p must be in the open interval (b, c). Since r-l(p) is in fo = [a, b], which is disjoint from (b, e), r-l(p) is not equal to p and so p can't have prime period n - 1. If the prime period of p were less than n - 1, then property (3) and the fact that p is not b or e would imply that the orbit of p is contained entirely in (b, e), and this would contradict property (4). So, p must have prime period n. Therefore, if a sequence of closed sets with the required properties exists for n, then there is a point p with prime period n .

3 The Topology of the Real Numbers The topology of a mathematical space is its structure or the characteristics it exhibits. In calculus, we were introduced to a few topological ideas, and we will need a few more in our study of dynamics. One of the fundamental questions of dynamics concerns the properties of the sequence x, f(x), P(x), P(x),.... To discuss these properties intelligently we need to understand convergence, accumulation points, open sets, closed sets, and dense subsets. In this section, we will limit our discussion to subsets of the real numbers; we will revisit the definitions when we introduce metric spaces in Chapter 11.

If a continuous function of the real numbers has a periodic point with prime period three, then it has a periodic point of each prime period. That is, for each natural number n there is a periodic point with prime period n . PROOF. Let {a, b, c} be a period three orbit of the continuous function f. Without loss of generality, we assume a < b < c. There are two cases: f(a) = b or f(a) = c. We suppose f(a) = b. This implies feb) = c and f(c) = a. The proof of the case f(a) = c is similar. Let 10 = [a, bJ and It = [b, cJ.

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