Fractals Northern llinois University分形北伊利诺斯大学.pptVIP

Fractals Compact Set Compact space X ? EN A collection {Ua; Ua ? EN} of open sets, X ? ? Ua. ? finite collection {Uak; k = 1 … n} Such that X ? ? Uak. Equivalent to every sequence of points in X has a subsequence that converges in X. Disconnected Applies to a subset S of a metric space X. Open sets U, V ? X S ? U ? V U ? V = ? {U, V} is a partition of S. Example: {0, 1} ? R Let U = (-0.5, 0.5) Let V = (0.5, 1.5) Connected 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. Example: [0, 1] is connected. Sketch proof by contradiction Let f: [0, 1] ? {0, 1} Assume continuous Suppose f(1) = 1 Let y be the least upper bound such that f(y) = 0 f is continuous, ? 1 ? x y, ? d |x – y| d, |f(x) – f(y)| 1. So f(x) = f(y) = 1. Path-Connected A space X is path-connected X is a metric space For any x, y ? X The function f: [0, 1] ? X f(0) = x, f(1) = y All path-connected spaces are connected. e.g. ellipse, disk, torus Not every connected space is path-connected. Example: Y = U ? V U = {(x,y): x = 0, -1 ? y ? 1} V = {(x,y): 0 x ? 1, y = sin(1/x)} Not path-connected If Y is disconnected U, V must be the partition. At the origin f(0,0) = 0 Neighborhood of the origin contains points in V. Cantor Set Subset of the interval [0, 1] At each step remove the open middle third of each interval. Continue ad infinitum. Set consists solely of disconnected points. The set is totally disconnected, but compact! C can be mapped onto [0,1]!! Countable Countable sets can be mapped into a subset of the natural numbers. N = {n ? Z: n 0} Can be finite or infinite Countable sets include: Empty set Finite sets Integers Rational numbers Uncountable sets cannot be mapped into N. Uncountable sets include: Real numbers Complex numbers Cantor set Contraction Map A map g is a contraction map Metric space X The function g: X ? X a ? [0,1] ? x1, x2 ? X d(g(x1), g(x2)) ? ad(x1, x2) Contract