# 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}, S_{0} **;**
S_{1}, S_{2}, ...

One can move it reversibly with respect to semicolon in both sides now.
We map this sequence into two real numbers

*x = 0. S*_{1} S_{2} S_{3} ...
y = 0. S_{0} 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.X*_{o}...X_{n} and *0.Y*_{o}...Y_{m}
where *n* and *m* are sufficiently large. Therefore periodic
sequence *(Y*_{m}...Y_{o}X_{o}...X_{n})
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.

Contents
Previous: Circle maps
Next: Smale horseshoe

*updated* 14 Nov 06