State of the System: The position of a
point on a circle. This position can be specified using the angle
made by a radius passing by the point and some direction of reference.
Dividing this angle expressed in degrees by 
, we obtain
a number x between
0 and 1. Since 
and are the same thing, 0 and 1 are identified.
For the same reason, if x is increased or decreased exactly by 1,2,3....
it does not change the state of the system. In the following, we
use the notation x Modulo 1 to say that we discard the integral part
of x. We also say, for instance, that 
.
Also, when we say that the distance between two points is d, we always mean
that we have taken the shortest path between these two points. For instance,
we say that the distance between 0.1 and 0.9 is 0.2 and not 0.8, the shortest
path consisting in going from 0.9 to 1 which is 0, and from 0 to 0.1.
Rule of Evolution:
For the uniform circular motion we have
,
where T
is the time taken for a complete revolution (often called the period).
The change in x is linear in t, in other
words x increases continuously at a fixed rate.
For the stroboscopic observations we have
where a is the time between two observations
divided by T.
Motivations: The second rule describes the positions, at successive and
equally separated instants, for instance 
of a point in uniform circular motion. The successive positions are then
, 
, 
....and can be
calculated using the continuous rule of evolution. We obtain that
. The idea of stroboscopic observation will
play an important role in the next example.
Main Results:
The evolution rule implies that 
.
If a is rational, the evolution is periodic. More
precisely, if 
,
then 
. For simplicity, we will assume that a is written in the
simplest way, i.e. that there are no integers that divide exactly both
and 
. For instance if a=0.4, we write 
and
not 
.
With this assumption,
it is possible to show
that during this cycle of length , all the values 
,
, 
,... 
are
only obtained once.
(The mathematically inclined might want to prove this statement
by first asuming that one of the possible values 
with 
,
appears twice in the first steps and that this assumption
leads to an inconsistency; consequently the assumption is wrong)
This is illustrated in Fig. 
for
a=23/33.
If a is irrational, the system never returns exactly to its initial state
after a finite number of steps. To see this suppose it returns to its
inatial value after n steps. This would mean that na=m for some integer
m. But this would mean that a is a rational number contrarily to
the hypothesis.
However, it will return as close as we want,
if we take enough steps.
We have seen in subsection ,
that an irrational number can be obtained as
the limit of a sequence of rational approximations.
If we consider better and better rational approximations for a, we
obtain longer and longer cycles.
As an example, if we take 
, we can follow the
Newton's method discussed previoulsy and calculate successive rational
approximations for a: 
. In this example the successive
cycles will be of length 2, 12, 408....
Taking the limit, we see that the cycle
becomes infinitely long and the intermediate values cover the interval [0,1]
uniformly. We call this type of behavior ergodic.
Poincaré Cycles.
This is a disgression for the mathematically inclined.
Instead of following the evolution of 1 point, we can consider the evolution
of all the points between 
and, say, 
.
The distance between
the points at the beginning and the end of the interval is 0.01. After one
step, the interval (denoted 
)
has evolved into all the points between 
and
which is also an interval of length 0.01. If we repeat the
evolution 100 times, we obtain 101 intervals of length 0.01 (including the
initial one). If we assume that all these intervals have no common parts,
we have to conclude that the total length covered by these intervals is
which is impossible since all the possible states
of the system form an interval of length one. So at least two intervals
must have a common part (which must be more than a single point).
Suppose that after inspection we discover that
the intervals 
and 
share
a subinterval of, say, length 0.002, then the distance between 
and
is less then 0.01, and so is the distance between 
and 
, 
and 
and so on (the translation by a
does not change the distance between two points).
These approximate cycles are called Poincaré cycles.
In general, we can use the same argument to show that after n steps,
at least one one of the 
, 
,.... 
will return at a distance
less than 1/n from .
Important Concepts: Cycles, periodicity, ergodicity.
Exercise : Describe as completely as possible
the evolution of the system (at
stroboscopic time) for 
and
a=3/10.
Calculate 
.
What would be if we had taken 
.
Which conclusions can you draw regarding small changes in the
initial conditions?
Exercise : Calculate , .... 
in the case and 
.
What is the smallest distance between two points of the
sequence and how many steps
are there between these two points. Can you understand this result from the
rational approximations of 
constructed in the previous Exercise
?
