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). |
Discontinuous baker's map is not very realistic. But it turn into continuous Smale horseshoe map if we fold stretched square as it 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. |