Foliations vs Laminations

Question

What's the big difference/similarity between foliations and laminations?

What kind of theorems hold for both of them?

Is there something that makes them essentially the same/different?

Answer 1

Not everybody agrees on what are the standard/canonical notions of foliation and lamination, but in the context of dynamical systems (still with many exceptions and/or schools), many people uses the first for something that can be parametrized more or less as in the flow box theorem, and uses the second when there are some possible topological problems.

Examples in this spirit would be the following.

Foliations: The collection of local stable/unstable manifolds for the geodesic flow on a compact manifold of strictly negative curvature. The same for an Anosov diffeomorphism or Axiom A diffeomorphism.

For example, the holonomies of a strong stable foliation inside the stable foliation are Lipschitz.

Laminations: The collection of global stable/unstable manifolds. The same for the local stable/unstable manifolds or the local unstable manifolds for the geodesic flow on a compact manifolds with nonzero Lyapunov exponents almost everywhere.

For example, the corresponding holonomies are only defined almost everywhere, and since now the size of the local manifolds is not necessarily bounded from below, something such as absolute continuity of the foliation/lamination may require additional assumptions. On the other hand, a version of the Lebesgue density for arbitrary $\sigma$-finite Borel measures is available, as for foliations, exactly with the same formulation.

But it is really impossible to make a summary of the differences and similarities. It really depends on what specific properties you may have in mind, and there are so many.