The sawtooth map is determined as
x
_{n+1} = 2x_{n} (mod 1)where x (mod 1) is the fractional part of x. In the binary
number system multiplying by 2 corresponds to the left shift by
one bit site and taking the fractional part corresponds to the upper bit
truncation. Therefore x is the Bernoulli shift of
_{n+1}x
_{n}x
_{o} = 0.01011 ...
x _{1} = 0.1011 ...
x _{2} = 0.011 ...and so on... The sequence (x
is called orbit of the point _{o }, x_{1} ...)x.
_{o} |

where

and

For even

Therefore there is infinite (countable) set of unstable periodic orbits and these orbits are dence in

In Fig.4 the second iteration of the map and two points of the period-2
orbit are shown. Period-2 orbit is obtaned from the symbolic sequence
σ= (01) (see Appendix)
x
_{0} = 0.0101... = 0.(01) = 1 / 11_{2}
= 1/3,
x _{1} = 0.1010... = 0.(10) =
10_{2} / 11_{2} = 2/3Two more combinations x and
_{2} = 0.(00) = 0x are fixed points of the map.
Two period-3 orbits are
_{3} = 0.(11) = 1x
_{0} = 0.001001... = 0.(001) =
1_{2} / 111_{2} = 1/7,
x _{1} = 0.(010) = 2/7,
x_{2} = 0.(100) = 4/7 |

However for given

For the sawtooth map after every iteration distance between close points
increases two times, thus a smooth density spreads uniformly two times too.
As since all points lay in the bounded *[0,1]* interval, therefore we
get uniform distribution of the points in the *n → ∞*
limit. This density is left unchanged by the sawtooth map (it is called
*stationary* or *invariant density*). Note that
points of an unstable periodic orbit make singular invariant density.

One can use this fact to substitute "time" average

In general case for a chaotic map

where

"wraps" regularly

Note, however that all points of the circle *[0,1]* are displaced by
the map on the same distance *Δ*. Therefore the distance between
two orbits is constant and density of any ensemble of points keeps its shape.
We have uniform invariant density with no mixing!

If invariant measure for a map is known then

E.g. for the degenerate circle map we have

where

For the sawtooth map correlation function is

Thus mixing leads to exponential decay of correlations for large

**Appendix**: Binary code conversion
to rational fraction

Consider decimal fraction at first

*0.(3) = 0.333... = 3 (10 ^{ -1} + 10^{ -2}
+ 10^{ -3} + ...) = 3/[10(1 - 1/10)] = 3/9 = 1/3*

(formula for the sum of geometric series is used). In a similar way for binary code

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