Third collection

5. Let p and q be relatively prime. A standard Seifert fibering of S^1\times D^2 corresponding to (p,q) is a decomposition of S^1\times D^2 into disjoint circles (fibers), constructed as follows. Starting with [0,1]\times D^2, decomposed into the segments [0,1]\times \{x\}, identify the disks \{0\}\times D^2 and \{1\}\times D^2 via a 2\pi p/q rotation. Note that the single segment [0,1]\times\{0\} becomes a (middle) fiber S^1\times\{0\}, while every other fiber in S^1\times D^2 is made from q segments [0,1]\times\{x\}. If q=1 the middle fiber is called ordinary, while if q>1 the middle fiber is called exceptional.

A manifold M 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 S^1\times D^2. 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 2-dimensional orbifold, denoted by B and called the base–also called the orbit surface–of the fibration. It has an underlying 2-dimensional surface B_0, but may have some special orbifold points corresponding to the exceptional fibers.

4. If smooth distribution is non-integrable, then it cannot be C^0 approximated by integrable ones.

Sergei Ivanov gave the following argument for exmaple (here): consider the following 2-dimensional distribution in \mathbb{R}^3: the plane at (x,y,z)\in\mathbb{R}^3 is spanned by vectors (1,0,0) and (0,1,x). Perturb this distribution within a small C^0 distance \epsilon\ll1. Consider the square in \mathbb{R}^2 with vertices (0,0), (1,0), (1,1), (0,1) and let \gamma be its boundary (counter-clockwise). This curve has a `lift’, which is a curve \tilde{\gamma} in \mathbb{R}^3 tangent to the distribution and projects to \gamma. 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 C^0. In the non-perturbed case, the unique lift ends at (0,0,1), hence in the perturbed case all lifts from (0,0,0) will end near (0,0,1). 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 \phi:\Sigma_A\to\mathbb{R} be a continuous function, L_\phi:C(\Sigma_A,\mathbb{R})\to C(\Sigma_A,\mathbb{R}) be a linear operator defined as L_\phi(f)(x)=\sum_{\sigma y=x}e^{\phi(y)}f(y).

Ruelle‘s Perron-Frobenius theorem: If \phi is Holder continuous, then there are a positive number \lambda>0, a positive function h\in C(\Sigma_A,\mathbb{R}) and a probability measure \nu\in\mathcal{M}(\Sigma_A) such that
L_\phi(h)=\lambda\cdot h, L_\phi^*\nu=\lambda\cdot \nu, \nu(h)=1 and the suprenorm

(*) \|\lambda^{-m}L^m_\phi g-\nu(g)\cdot h\|\to 0 for all Holder continuous function g:\Sigma_A\to\mathbb{R}.

Moreover the pressure P(\sigma,\phi)=\log\lambda and \mu=h\cdot\nu is the unique equilibrium state (or Gibbs measure) of \phi.

A priori, the existence of \nu and h follow from Schauder–Tychonoff theorem (so the uniqueness is not automatically guaranteed). The last characterization (*) gives us the uniqueness of the pari (\nu,h): if there is another pair (\mu,f), then \mu(g)\cdot f=\nu(g)\cdot h for all Holder g. Since both \mu and \nu are probabilities, \mu(1)=\nu(1)=1 and hence f=h. Now \mu=\nu since \mu(g)=\nu(g) for all Holder g.

2. Let [0,1] be the unit interval equiped with Lebesgue measure m, (\{0,1\}^{\mathbb{N}},\sigma) be the full-shift and \mu_p=(p,1-p)^{\mathbb{N}} be the standard ergodic measure for all p\in(0,1). Let \pi:\{0,1\}^{\mathbb{N}}\to[0,1] be the standard binary representation.

Since \pi is almost one-to-one and \mathrm{supp}(\mu_p)=\{0,1\}^{\mathbb{N}} for each p\in(0,1), we see \mathrm{supp}(m_p)=[0,1]. But we also know \pi(\mu_{0.5})=m and all other m_p=\pi(\mu_p) are singular with respect to m. So m_p (for each p\in(0,0.5)) is a singular continuous measure whose support is all of [0,1].

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

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