## 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.