Can one tell true randomness apart from the output of a deterministic (chaotic) system?

According to several theorems of Ornstein in ergodic theory, there is
no way to tell between true randomness (i.e. an assumed a priori
distribution with an unknown mechanism) and deterministic chaos.

This is the idea.  Consider a billiard ball (idealized as a point)
moving on a rectangular table which has a convex obstacle in the middle
of it.  This system is known to be sensitive to initial conditions.
Now start the ball rolling on the table.  You do the following
observation on the ball:

Run a line half way down the table so that it is separated into a left
side and a right side.  Now every minute look to see if the ball is in
the right side or if it is the left side.  Write an H if it is in the
right side and a T if it is the left hand side.  Ornstein showed the
following:

a) For almost all such trajectories of the billiard ball the sequence
   of H and T's is random.

b) There exists another (extremely complicated) deterministic way to
   partition the table into two sets, such that the analogue of the
   above (calling H if it is in one of the sets, T in the other) gives
   you for almost all trajectories a sequence of H's and T's that is
   stochastically independent with probability 1/2 of getting H (i.e.
   is exactly a sequence of coin flips).

Thus from b) it is seen that one cannot distinguish between randomness
coming from a deterministic system and one coming from a realization of
an a priori random system.

Where is the randomness coming from in the deterministic system?  The
point is that when you do a finite precision measurement on the
continuous system, you are creating a discrete stochastic process, the
randomness came from having to pick a random trajectory (recall the
result was for *almost all* trajectories).  This may be confusing but
here is a simpler example of how a priori random can come from a
deterministic system:

  Consider the map f(x)={2x} on (0,1) i.e. x goes to the fractional
  part of 2x.  The system we look at is the one that iterates f every
  second i.e. f_{t+1}(x)=f (f_t(x)).  This is clearly a deterministic
  system.  Split up (0,1) into two sets A=(0,1/2), B=(1/2,1).  We do
  the analogue of the above experiment: every second we check if the
  point is in B then write H, or if it is in A then write T.

Now for almost any real 0<x<1 its binary expansion is a stochastically
independent sequence of 0's and 1's.  What we get as our list now is
just the binary expansion of the x we started with.  So once again
randomness comes out of a deterministic system (though not from a
physical one obeying Newton's laws like billiards).

The construction of Ornstein's partition is part of his Isomorphism
Theorems, and the fact that any partition gives you randomness follows
from the factor theorem of Isomorphism theory [Ergodic Theory,
Randomness and Dynamical Systems, Yale University Press, 1972]

[ Editor: the posting is unsigned in the collected material. ]