新編文檔-Fractals-NorthernIllinoisUniversity分形-北伊利諾斯大學(xué)-精品文檔

1、FractalsCompact Set舉 Compact space Xcz ENA collection U; UuEN of open sets, Xu u 113 finite collection U,; k = 1 .nSuch that Xu u 11於舉 Equivale nt to every seque nee of points in Xhas a subsequence that converges in X.A space Xis compact if and only if it is closed and bounded.鬱 Applies to a subset

2、Sof a metric space X. Open sets U /u X SuU* Ur V=0鶴U, Vs a partition of S.鶴 Example: 0, 1 u R rLet U= (-0.5, 0.5)Let V= (0.5, 1,5)Connected鞭 A space is disconnected if and only if there is a continuous map onto 0, 1= If a space has no partition it is connected.i.e. if its not disconnected.題 Example:

3、 0, 1 is connected.題 Sketch proof by contradict!on Let/:O,1 Assume continuous Suppose f(l) = 1 Let y be the least upper bound such that/(y) = 0 /is conti nu ous, 3 l>x>y, 3 5lx-yl<8, f(x)心)lvl So夬x)二心)二 1.Path-Connected題 Example: Y= V( U= (x,y):x = 0? -1 < y < 1 V= (x,y)： 0<x< l

4、,y = sin(l/x) Not path-connected舉 If Yis disconnected U, V must be the partition. At the origin 7(0,0) = 0 Neighborhood of the origin contains points in V.A space Xis path-connectedI Xis a metric spaceFor any x, y e XE The function/: 0,1 t XXo)= x,Ai) = yAll path-connected spaces are connected.e.g.

5、ellipse, disk, torusNot every connected space is pathconnected.Cantor SetSubset of the interval 0, 1E At each step remove the open middle third of each interval.E Continue ad infinitum.H Set consists solely of disc on nected points.e The set is totallydisc on nected, but compact!00舉 Ccan be mapped o

6、nto 0,1!鞭 Countable sets can be mapped into a subset of the natural numbers.N = /7 e 乙 n > 0Can be finite or infinite觀 Countable sets in elude: Empty setFinite setsIn tegers Rational numbersUn countable sets cannot be mapped into N.Uncoun table sets in elude:Real numbers B Complex numbersCantor s

7、etContraction Map賤 Suppose for 1 < n < M gn is a contraction map鞭 A map g is a contraction mapMetric space XThe function g： Xt X a e 0,1V x2 e Xd(g(“)，g(£) < adg x2)鞭 Contract!on maps have a fixed point: g(x) = xEG(A) = gn(A):l<n<Na con traction map on H(X)Koch Curve鞭 Start on th

8、e interval 0, 1At each step remove the open middle third of each interval and add two equal segments. Continue ad infinitum. Set is connected.鞭 The contraction map is fractal.A fractal curveSelf-similar at many scales舉 For Euclidean spaceE1 is 1- dimesi onalE2 is 2-dimesionalE3 is 3-dimesionalIndln（Qo/a）Exp orient sugge