In dynamical systems theory, the horseshoe map was introduced by Stephen Smale as a simple model of complex behavior. It is given on the unit square by the formula:

(1)     <math>(x_{n+1},y_{n+1}) = H((x_n,y_n))<math>,

where:

(2)     <math>H(x,y)=\left\{\begin{matrix} (2x,\frac {y} {2}), & \mbox{if }x< \frac {1} {2} \\ (2x-1,1-\frac {y} {2}), & \mbox{if }x\ge\frac {1} {2} \end{matrix}\right.

<math>

This map serves as a model for general behavior at transverse homoclinic points, and can be fairly easily shown to have an invariant compact set on which it acts as a shift map.

Using a few hundred mirrors, one can build an optical universal Turing machine in one's backyard, using the Horseshoe map.