Any bounded (holomorphic) entire function is constant.

In fact, considerably more than this is known: the real and imaginary parts of a holomorphic function are harmonic functions; not only do no bounded harmonic functions exist in R^n, but in fact minimal rates of growth may be established.

Actually, proving a discrete version of this theorem is rather more difficult than the continuous versions -- see Z^n admits no bounded harmonic function for more details.

Liouville's theorem in Physics refers to the theorem that under Hamiltonian evolution the volume in phase space occupied by a collection of particles remains constant - that is these particles behave like an incompressible fluid.

For example consider a box full of monoatomic gas molecules. The phase space for this system is 6 dimensional. Now each molecule may be represented by a point in phase space. Liouville's theorem states that as the system evolves the total volume occupied by these particles in phase space will be conserved.

The Liouville Theorem is in fact quite easy to understand ... though a formal proof might be more complicated.

Anyway. Physicists like to look at problems in phase space, that is, a coordinate system of spatial and momentum coordinates. In that space an object therefore usually has 6 coordinates - 3 for space (x1,x2,x3) and 3 for momentum (p1,p2,p3). These values determine the future of the object - using the Hamilton formalism one can calculate its trajectory in phase space.

Now suppose you have a number of points in phase space like this:

p /\        .
  |        ....
  |        ......
  |      .....
  |       ..

If now one of the inner points' trajectories were to touch one of the outer points', from then on it would evolve right along with that other point. That means it can not cross! And in fact, two trajectories cannot even touch because there are always other trajectories in between (the coordinates are real numbers) which would have to be crossed first!

Therefore an inner point will always remain an inner point ... the group of points may change its shape, but the area (or volume in 6d) has to stay constant because there's no place to go for the inner points. A phase space volume behaves like an incompressible liquid and the phase space density is constant!

PS: It seems that the theorem ariels refers to is a different one, also called Liouville's Boundedness Theorem.

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