Fractals

Compact Set  Compact space X  E N A collection {U  ; U   E N } of open sets, X   U .A collection {U  ; U   E N } of open sets, X   U .  finite collection {U  k ; k = 1 … n }  finite collection {U  k ; k = 1 … n } Such that X   U  k.Such that X   U  k.  Equivalent to every sequence of points in X has a subsequence that converges in X. y1y1 X Y ynyn p A space X is compact if and only if it is closed and bounded.

Disconnected  Applies to a subset S of a metric space X. Open sets U, V  XOpen sets U, V  X S  U  VS  U  V U  V = U  V =   {U, V} is a partition of S.  Example: { 0, 1 }  R Let U = (-0.5, 0.5)Let U = (-0.5, 0.5) Let V = (0.5, 1.5)Let V = (0.5, 1.5) S S V U

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.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,   |x – y| < , |f(x) – f(y)| < . So f(x) = f(y) = 1.

Path-Connected  A space X is path-connected X is a metric spaceX is a metric space For any x, y  XFor any x, y  X The function f : [0, 1]  XThe function f : [0, 1]  X f(0) = x, f(1) = yf(0) = x, f(1) = y  All path-connected spaces are connected. e.g. ellipse, disk, toruse.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.At each step remove the open middle third of each interval. Continue ad infinitum.Continue ad infinitum. Set consists solely of disconnected points.Set consists solely of disconnected points.  The set is totally disconnected, but compact!  C can be mapped onto [0,1]!! 0  1 2 3

Countable  Countable sets can be mapped into a subset of the natural numbers. N = { n  Z : n > 0}N = { n  Z : n > 0} Can be finite or infiniteCan be finite or infinite  Countable sets include: Empty setEmpty set Finite setsFinite sets IntegersIntegers Rational numbersRational 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 XMetric space X The function g : X  XThe function g : X  X a  [0,1]a  [0,1]  x 1, x 2  X  x 1, x 2  X d(g(x 1 ), g(x 2 ))  ad(x 1, x 2 )d(g(x 1 ), g(x 2 ))  ad(x 1, x 2 )  Contraction maps have a fixed point: g(x) = x. g X x  Suppose for 1  n  N, g n is a contraction map.  G : H ( X )  H ( X ) G(A) =  { g n (A): 1  n  N } a contraction map on H(X)

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

Fractal Dimension  For Euclidean space E 1 is 1-dimesionalE 1 is 1-dimesional E 2 is 2-dimesionalE 2 is 2-dimesional E 3 is 3-dimesionalE 3 is 3-dimesional  Exponent suggests a logarithm. Take a unit and divide into equal subdivisions a << a 0Take a unit and divide into equal subdivisions a << a 0   Apply to the Cantor set 2 n segments next a0a0 a