Smale horseshoe and 2D strange repellers

2D Cantor repeller

It is not difficult to get generalized baker's map if we stretch unit square 3 times, cut it into 3 pieces and put them one on another. Dynamics of this mapping is equivalent to the Bernoulli shifts of ternary symbols 0, 1 and 2. To get 2D Cantor repeller in every iterations we throw away the central 1/3 interval. Under iterations almost all points eventually leave the square. Remaining invariant set has zero measure. After the first iteration points of the set are into intersection of two horizontal strips 0, 2 and two vertical strips 0 -1, 2 -1 (they are preimages of 0, 2). After the second iteration invariant set is situated into intersection of the four horizontal (blue) and four vertical (yellow) strips. In the limit we get 2D Cantor set. Points of the set have coordinates
    x = 0. S1 S2 S3 ...     y = 0. S0 S-1 S-2 ... ,
where Sk = 0 or 2. Dynamics on this invariant set is equivalent to the Bernoulli shifts of the string
    ... S-2 S-1 S0 ; S1 S2 ...

Smale horseshoe map

Discontinuous baker's map is not very realistic. But it turn into continuous Smale horseshoe map if we fold stretched square as it is shown to the left. It differs from the generalized baker's map in orientation of the third piece. Its invariant set is again fractal repeller with zero measure. It has countable set of unstable periodic cycles and continuum of chaotic orbits. If a map has horseshoe than it has a region (possibly repelling) with complex chaotic dynamics. We meet horseshoe in the Henon map further.
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updated 29 June 2007