Smale Horseshoe Map
The basic topological operations for constructing an Attractor consist of stretching (which gives sensitivity toinitial conditions) and folding (which gives the attraction). Since trajectories in Phase Space cannot cross, therepeated stretching and folding operations result in an object of great topological complexity.
The Smale horseshoe map consists of a sequence of operations on the unit square. First, stretch by a factor of 2 in the 
direction, then compress by 
in the 
direction. Then, fold the rectangle and fit it back into the square. Repeatingthis generates the horseshoe attractor. If one looks at a cross-section of the final structure, it is seen to correspond to aCantor Set.
References
Gleick, J. Chaos: Making a New Science. New York: Penguin, pp. 50-51, 1988. Rasband, S. N. Chaotic Dynamics of Nonlinear Systems. New York: Wiley, p. 77, 1990. Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.
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