5. Let and be relatively prime. A standard Seifert fibering of corresponding to is a decomposition of into disjoint circles (fibers), constructed as follows. Starting with , decomposed into the segments , identify the disks and via a rotation. Note that the single segment becomes a (middle) fiber , while every other fiber in is made from segments . If the middle fiber is called ordinary, while if the middle fiber is called exceptional.

A manifold is a Seifert manifold if it admits a decomposition into disjoint circles (fibers), such that each fiber has a neighborhood fibers-preservingly diffeomorphic to a neighborhood of a fiber in some standard Seifert fibering of . A compact Seifert fiber space has only a finite number of exceptional fibers. A Seifert manifold is often allowed to have a boundary (also fibered by circles, so it is a union of tori). Any 3-manifold supporting a foliation by circles is Seifert.

The set of fibers forms a -dimensional orbifold, denoted by and called the base–also called the orbit surface–of the fibration. It has an underlying -dimensional surface , but may have some special orbifold points corresponding to the exceptional fibers.

4. If smooth distribution is non-integrable, then it cannot be approximated by integrable ones.

Sergei Ivanov gave the following argument for exmaple (here): consider the following 2-dimensional distribution in : the plane at is spanned by vectors and . Perturb this distribution within a small distance . Consider the square in with vertices , , , and let be its boundary (counter-clockwise). This curve has a `lift’, which is a curve in tangent to the distribution and projects to . The lift is found by solving an o.d.e., so it is unique if the distribution is smooth but may be non-unique if it is only . In the non-perturbed case, the unique lift ends at , hence in the perturbed case all lifts from will end near . This implies that the distribution is not integrable: if it was integrable, there would be at least one lift (the one contained in a leaf of a foliation) that ends near the origin.

The proof in the general case is similar.

3. Let be a continuous function, be a linear operator defined as .

**Ruelle**‘s *Perron-Frobenius theorem*: If is Holder continuous, then there are a positive number , a positive function and a probability measure such that

, , and the suprenorm

(*) for all Holder continuous function .

Moreover the pressure and is the unique equilibrium state (or Gibbs measure) of .

A priori, the existence of and follow from Schauder–Tychonoff theorem (so the uniqueness is not automatically guaranteed). The last characterization (*) gives us the uniqueness of the pari : if there is another pair , then for all Holder . Since both and are probabilities, and hence . Now since for all Holder .

2. Let be the unit interval equiped with Lebesgue measure , be the full-shift and be the standard ergodic measure for all . Let be the standard binary representation.

Since is almost one-to-one and for each , we see . But we also know and all other are singular with respect to . So (for each ) is a singular continuous measure whose support is all of .

1. By the local genericity of the universal dynamics at a wild homoclinic class, any diffeomorphism having a wild homoclinic class can be –approximated by a diffeomorphism with a hyperbolic basic set that exhibits –persistent homoclinic tangency.