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Fractals

Fractals. Compact Set. Compact space X  E N A collection {U a ; U a  E N } of open sets, X   U a .  finite collection {U a k ; k = 1 … n } Such that X   U a k . Equivalent to every sequence of points in X has a subsequence that converges in X. X. y n. y 1. p. Y.

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Fractals

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  1. Fractals

  2. 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. X yn y1 p Y A space X is compact if and only if it is closed and bounded.

  3. 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) Disconnected S U S V

  4. 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 Letf: [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. Connected

  5. 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. Path-Connected

  6. 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]!! Cantor Set 0 1 2 3 

  7. 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 Countable

  8. 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) Contraction maps have a fixed point: g(x) = x. Suppose for 1 nN, gn is a contraction map. G: H(X) H(X) G(A) =  {gn(A): 1 nN} a contraction map on H(X) Contraction Map X g x

  9. Start on the interval [0, 1] At each step remove the open middle third of each interval and add two equal segments. Continue ad infinitum. Set is connected. The contraction map is fractal. A fractal curve Self-similar at many scales Koch Curve 0 1 2 5

  10. For Euclidean space E1 is 1-dimesional E2 is 2-dimesional E3 is 3-dimesional Exponent suggests a logarithm. Take a unit and divide into equal subdivisions a << a0 Apply to the Cantor set Fractal Dimension a a0 2n segments next

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