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3.5] What is a Cantor set? (nonlinear science)




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This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

3.5] What is a Cantor set? (nonlinear science)

(Thanks to Pavel Pokorny for contributing to this answer)

A Cantor set is a surprising set of points that is both infinite (uncountably
so, see [2.14]) and yet diffuse. It is a simple example of a fractal, and
occurs, for example as the strange repellor in the logistic map (see [2.15])
when r>4. The standard example of a Cantor set is the "middle thirds" set
constructed on the interval between 0 and 1. First, remove the middle third.
Two intervals remain, each one of length one third. From each remaining
interval remove the middle third. Repeat the last step infinitely many times.
What remains is a Cantor set.

More generally (and abstrusely) a Cantor set is defined topologically as a
nonempty, compact set which is perfect (every point is a limit point) and
totally disconnected (every pair of points in the set are contained in
disjoint covering neighborhoods).

See also
http://www.shu.edu/html/teaching/math/reals/topo/defs/cantor.html
http://personal.bgsu.edu/~carother/cantor/Cantor1.html
http://mizar.uwb.edu.pl/JFM/Vol7/cantor_1.html

Georg Ferdinand Ludwig Philipp Cantor was born 3 March 1845 in St Petersburg,
Russia, and died 6 Jan 1918 in Halle, Germany. To learn more about him see:
http://turnbull.dcs.st-and.ac.uk/history/Mathematicians/Cantor.html
http://www.shu.edu/html/teaching/math/reals/history/cantor.html

To read more about the Cantor function (a function that is continuous,
differentiable, increasing, non-constant, with a derivative that is zero
everywhere except on a set with length zero) see
http://www.shu.edu/projects/reals/cont/fp_cantr.html

 

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