Baker's map

Chaotic 1D sawtooth map is irreversible due to upper bit truncation under Bernoulli shifts. To get reversible map let us take a binary symbolic sequence infinite in both sides
    ... S-2, S-1, S0 ; S1, S2, ...
One can move it reversibly with respect to semicolon in both sides now. We map this sequence into two real numbers
    x = 0. S1 S2 S3 ...     y = 0. S0 S-1 S-2 ...
It is evident that after the left shift of the symbolic sequence we get new values
    x' = 2x (mod 1)
    y' = 1/2 (y + [2x])
where [x] is the integer part x. This map of unit square into itself is called the baker's map. The square is squeezed uniformly two times in vertical direction and stretched in horizontal direction. Then "baker" cuts the right half and put it over the left one. The result of two first iterations is shown in Fig.1 below.

Unstable periodic orbits

As like as for the sawtouth map periodic orbits can be found from symbolic sequences. So symbolic sequences (0) and (1) correspond to fixed points (x, y) = (0, 0) and (1, 1). Periodic sequence (10) corresponds to the period-2 orbit {(1/3, 2/3), (2/3, 1/3)}. And out of ...001;001... we get {(1/7, 4/7), (2/7, 2/7), (4/7, 1/7)} orbit and out of ...011;011... we get {(3/7, 6/7), (6/7, 3/7), (5/7, 5/7)} one.

Any x and y can be approximated arbitrary close by 0.Xo...Xn and 0.Yo...Ym where n and m are sufficiently large. Therefore periodic sequence (Ym...YoXo...Xn) goes arbitrary close to any point of unit square. So periodic orbits make a dense set on it.

Mixing and chaotic orbits

Due to stretching in the horizontal direction all close points diverge exponentially under iterations. As like as for the sawtouth map random symbolic sequence goes arbitrary close to any point of square (ergodicity).

It is evident that under baker's iterations any region turns into a set of narrow horizontal strips. Eventually it fills uniformly all unit square (mixing). Reverse iterations turn the region into narrow vertical strips and mix it all the same.

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updated 14 Nov 06