3 lots for lot's

First of all Poincare's Recurrence Theorem is important historically because it poses a serious obstacle to modeling an ideal gas in a box as a mechanical system consisting of hard spheres bouncing around. See The wikipedia article on the second law of thermodynamics but the basic idea is that such a mechanical system preserves a natural volume on its phase space (Liouville's theorem) but this is in contradiction (by Poincare's Theorem) with the fact that entropy decreases along all trajectories (assuming that entropy is a continuous function in phase space).

Also, I like this proof that any harmonic function of a compact Riemannian manifold is constant:

Take the gradient flow of your harmonic function and notice that it's volume preserving (more or less by the definition of harmonicity since divergence of the gradient is zero). Now apply Poincare's Recurrence Theorem so that you now have a dense set of recurrent orbits. However since this is a gradient flow, the value of the function decreases strictly along any non-singular orbit, this means all the recurrent orbits are singular so that the gradient has a dense set of zeros. Conclusion: the function is constant.

Last but not least Poincare's Recurrence Theorem is more or less just the Pigeonhole Principle (finite space + lot's lots of stuff = lot's lots of overlap) which is certainly a fundamental mathematical fact even if not so many theorems follow directly from it. See Terrence Tao's course notes on Ergodic Theory where he emphasized this point very beautifully.

2 typo

First of all Poincare's Recurrence Theorem is important historically because it poses a serious obstacle to modeling an ideal gas in a box as a mechanical system consisting of hard spheres bouncing around. See The wikipedia article on the second law of thermodynamics but the basic idea is that such a mechanical system preserves a natural volume on its phase space (Liouville's theorem) but this is in contradiction (by Poincare's Theorem) with the fact that entropy decreases along all trajectories (assuming that entropy is a continuous function in phase space).

Also, I like this proof that any harmonic function of a compact Riemannian manifold is constant:

Take the gradient flow of you're your harmonic function and notice that it's volume preserving (more or less by the definition of harmonicity since divergence of the gradient is zero). Now apply Poincare's Recurrence Theorem so that you now have a dense set of recurrent orbits. However since this is a gradient flow, the value of the function decreases strictly along any non-singular orbit, this means all the recurrent orbits are singular so that the gradient has a dense set of zeros. Conclusion: the function is constant.

Last but not least Poincare's Recurrence Theorem is more or less just the Pigeonhole Principle (finite space + lot's of stuff = lot's of overlap) which is certainly a fundamental mathematical fact even if not so many theorems follow directly from it. See Terrence Tao's course notes on Ergodic Theory where he emphasized this point very beautifully.

1

First of all Poincare's Recurrence Theorem is important historically because it poses a serious obstacle to modeling an ideal gas in a box as a mechanical system consisting of hard spheres bouncing around. See The wikipedia article on the second law of thermodynamics but the basic idea is that such a mechanical system preserves a natural volume on its phase space (Liouville's theorem) but this is in contradiction (by Poincare's Theorem) with the fact that entropy decreases along all trajectories (assuming that entropy is a continuous function in phase space).

Also, I like this proof that any harmonic function of a compact Riemannian manifold is constant:

Take the gradient flow of you're harmonic function and notice that it's volume preserving (more or less by the definition of harmonicity since divergence of the gradient is zero). Now apply Poincare's Recurrence Theorem so that you now have a dense set of recurrent orbits. However since this is a gradient flow, the value of the function decreases strictly along any non-singular orbit, this means all the recurrent orbits are singular so that the gradient has a dense set of zeros. Conclusion: the function is constant.

Last but not least Poincare's Recurrence Theorem is more or less just the Pigeonhole Principle (finite space + lot's of stuff = lot's of overlap) which is certainly a fundamental mathematical fact even if not so many theorems follow directly from it. See Terrence Tao's course notes on Ergodic Theory where he emphasized this point very beautifully.