## Collections again

8. Definition. Given a family of maps $T_\epsilon:X\to X$ with corresponding invariant densities $\phi_\epsilon$. Then $T_0$ is said to be acim-stable if $lim_{\epsilon\to 0}T_\epsilon=T_0$ implies $lim_{\epsilon\to 0}\phi_\epsilon=\phi_0$.
The limits are taken with respect to properly chosen metrics on the space of maps and densities, respectively.

Functions of the bounded variation are continuous except at a most countable number of points, at which they have two one-sided limits.

7. Let $\mathcal{H}=(\mathbb{R}^3,\ast)$ be the 3D Heisenberg group, with $(a,b,c)\ast(x,y,z)=(a+x,b+y,c+z+ay)$. Let $\Gamma=\langle\alpha,\beta,\gamma|\alpha\ast\beta=\beta\ast\alpha\ast\gamma,\alpha\ast\gamma=\gamma\ast\alpha,\beta\ast\gamma=\gamma\ast\beta\rangle$ be a cocompact discrete subgroup (for example $\mathbb{Z}^3=\langle \mathbf{i},\mathbf{j},\mathbf{k}\rangle$). Then $M=\mathcal{H}/\Gamma$ is a 3D nilmanifold. A general non-toral
three-dimensional nilmanifold is also of this form. Suppose we have a homomorphism $h:\mathcal{H}\to\mathbb{R}$, which is of the form $(x,y,z)\mapsto ax+by$, which induces a 2D-foliation, say $\mathcal{F}_h$ on $\mathcal{H}$ and on $M$.

Theorem. Every Reebless foliation on $M$ is almost aligned with some $\mathcal{F}_h$.
Plante for $C^2$, Hammerlindl and Potrie for $C^{1,0}$.

Theorem. Every partially hyperbolic system on $M$ is accessible.
J. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures (convervative), Hammerlindl and Potrie (general)

6. Let $r\ge 1$ and $S:\mathbb{R}^{r+1}\to\mathbb{R}$ be a $C^2$ function. Consider the solutions $x:\mathbb{Z}\to \mathbb{R}$ of the recurrence relation:
($\ast$) $\displaystyle R(x_{i-r},\cdots,x_{i+r}):=\sum_j \partial_{x_i}S(x_j,\cdots,x_{j+r})=0,$ for all $i\in\mathbb{Z}$.
Note that ($\ast$) is actually a finite sum of $r+1$ terms over $j=i-r,\cdots,i$. It is the derivative of formal series $W(x)=\sum_j S(x_j,\cdots,x_{j+r})$ with respect to $\partial_{x_i}$.
Example. Billiards, or generally twist maps, where $r=1$ and $S$ is the generating function, the solution gives the configuration of an orbit.

There are some conditions:
Periodicity. $S(x+1)=S(x)$. So $S$ descends to a map on $\mathbb{R}^{\mathbb{Z}}/\mathbb{Z}$.
Monotone. $\displaystyle\partial_{x_i,x_k}S(x_j,\cdot,x_{j+r})\le 0$ for all $j$ and all $i\neq k$, and $\displaystyle\partial_{x_j,x_{j+1}}S(x_j,\cdot,x_{j+r}) < 0$ for all $j$.
Coercivity. $S$ is bounded from below and there exists $k$ such that $S(x_j,\cdots,x_{j+r})\to\infty$ as $|x_k-x_{k+1}|\to\infty$.
Under these conditions the ($\ast$) is called a monotone variational recurrence relation.

A sequence $x$ is said to be a global minimizer, if $W(x)\le W(x+v)$ (understand as over all intervals) for all sequences $v$. Clearly a global minimizer solves ($\ast$). The collection of global minimizers is also closed under coordinately convergence.

For a real number $a$, a sequence $x$ with $x_0=a$ is called an $a$-minimizer, if it is minimizes among all $y$‘s with $y_0=a$.
Ana-minimizers in general need not be solutions to ($\ast$).
Given a rational $p/q$, we consider the operator $\tau_{p,q}$ (shift $p$ and subtract $q$) and Birkhoff orbits of rotational number $p/q$ prime sum $W_{p,q}=S(x_0,\cdots, x_{r})+\cdots+S(x_{p-1},\cdots,x_{p-1+r})$ over the periodic ones $x=\tau_{p,q}(x)$.

Periodic Peierls barrier. Let $a$ be a real and $p,q$ be coprime. Then as
$\displaystyle P_{p,q}(a):= \min_{\tau_{p,q}x=x,x_0=a} W_{p,q}(x)-\min_{\tau_{p,q}x=x}W_{p,q}(x)$.

It is easy to see that
There exists a periodic minimizer $x\in M_{p,q}$ with $x_0 =a$ if and only if $P_{p,q}(a)=0$.

$M_{p,q}$ gives an invariant curve if and only if $P_{p,q}(\cdot)\equiv 0$.

Then the Peierls barrier at a general frequency is defined as $P_{\omega}(a)=\lim_{p/q\to\omega}P_{p,q}(a)$ when the limit exists (see Mramor and Rink, arxiv:1308.3073).

5. Let $T:A\to A=\mathbb{R}\times [0,1]$ be a monotone twist map, $h$ be the generating function of $T$: $T(x,y)=(X,Y)$ iff $y=-h_1(x,X)$ and $Y=h_2(x,X)$. Then the twist condition $\frac{\partial X}{\partial y}>0$ is equivalent to $h_{12}<0$.
Let's further suppose $h_{12}<-c$. A sequence $\{x_n\}$ corresponding to the orbit $(x_n,y_n)=T^n(x_0,y_0)$ is called the configuration. In particular we have
($\ast$) $h_2(x_{n-1},x_n)+h_1(x_n,x_{n+1})=0$.

Consider a $C^1$-family $\{f_n(t):0\le t\le 1\}$ of the configurations. Then taking derivative of $t$ with respect to ($\ast$), we get
($\star$) $h_{12}\dot x_{n-1}+h_{22}\dot x_n+h_{11}\dot x_n+h_{12}\dot x_{n+1}=0$.

Definition. A sequence $\{\xi_n\}$ is called a Jacobi field along $\{x_n\}$ if it satisfies the equation
$h_{12}(x_{n-1},x_n)\xi_{n-1}+(h_{22}(x_{n-1},x_n)+h_{11}(x_n,x_{n+1}))\xi_n+h_{12}(x_n,x_{n+1})\xi_{n+1}=0$
Then $\{x_n\}$ is said to have conjugacy, if there exists a nonzero Jacobian field $\{\xi_n\}$ along $\{x_n\}$ vanishing at some $m < n$.
The map $T$ is said to be conjugacy-free, if there is no configuration with conjugacy.

Theorem. (J. CHENG and Y. Sun, 1995) Every conjugacy-free monotone twist map is integrable, and the phase space is foliated by non-null invariant curves.
The converse is also true. The intermediate property is that the Hassian of the action function on each segment of the configuration of each orbit is positively definite.

Lemma. Suppose $T$ is conjugacy-free. Then any two configurations $\{x_n^a\}$ and $\{x_n^b\}$ cross at most once.
Proof. Let’s assume they already cross once at $i=-1,0$: $x_{-1}^a \le x_{-1}^b$ and $x_{0}^a> x_{0}^b.$ Pick two linear functions $f_i(t)$ connecting $x^a_i$ and $x^b_i$, $i=0,1$. Then we can solve $f_i(t)$ from ($\ast$) connecting $x^i_a$ and $x^i_b$ for all other $i$. Clearly $f_{-1}'(t) \ge 0$ and $f_{0}'(t) < 0$. If they cross again after $i\ge 2$, we see that $f_{i}'(t) \ge 0$, and the positive definition would force $f_{0}'(t) \ge 0$, contradiction.

4. Let $X$ be a metric space and $U\subset X$ be a connected open subset, whose $\epsilon$-kernel is defined as $U_\epsilon=\{x\in U:d(x,\partial U)>\epsilon\}$. Clearly $U_\epsilon\nearrow U$ as $\epsilon\to 0$. But $U_\epsilon$ may not be connected, since there might some narrow channels connecting $U$. Let $\epsilon_0>0$ be small enough such that $p\in U_{\epsilon_0}\neq\emptyset$. Then for all $\epsilon<\epsilon_0$, let $U^1_\epsilon$ be the connect component of $U_\epsilon$ containing the marked point $p$.

Proposition. $U^1_+:=\bigcup_{\epsilon>0}U^1_\epsilon= U$.
Proof. It suffices to show that $U^1_+$ is closed in $U$. So pick a point $x\in U\cap \overline{U^1_+}$. Let $\epsilon>0$ such that $B(x,\epsilon)\subset U$, and $\delta<\epsilon/3$ such that $U^1_\delta\cap B(x,\epsilon/3)\neq\emptyset$. Then we actually have $x\in U^1_\delta$. This ends the proof.

3. Let $M$ be a closed manifold (mostly 3 dimension) and $\mathcal{F}$ a foliation on $M$. A leaf $F\in\mathcal{F}$ is closed if $\overline{F}=F$ (like a periodic orbit or a genus-$g$ surface). It is proper if $\overline{F}\backslash F$ is closed (like the graph of $y=\sin(1/x)$). It is recurrent if it is either closed (trivial proper) or non-proper (like a line on $\mathbb{T}^2$ with irrational slope). Let $\mathcal{C}$ be the part of closed leaves (note that $\mathcal{C}$ may not be closed, like the periodic orbits of Anosov flow). Let $\mathcal{P}$ be the part of proper leaves (nontrivial proper: not closed) . In particular all leaves outside $\mathcal{P}$ are recurrent.

The foliation $\mathcal{F}$ is said to be non-wandering if $\mathcal{P}$ has no interior. And $\mathcal{F}$ is said to be recurrent if $\mathcal{P}=\emptyset$, that is, every leaf of $\mathcal{F}$ is recurrent. Moreover, $\mathcal{F}$ is said to be almost periodic, if $\{\overline{F}:F\in\mathcal{F}\}$ forms a new decomposition of $M$ (either disjoint or coincide). Yokoyama observed the following proposition:

Proposition. A almost periodic foliation is recurrent.
Proof. Let $F\in\mathcal{F}$ be a non-closed element and $x\in\overline{F}\backslash F$. Clearly $F(x)\cap F=\emptyset$. Then almost periodicity implies that $\overline{F{x}}=\overline{F}$. So $\overline{F}\backslash F=\overline{F{x}}\backslash F\supset F(x)\neq\emptyset$. So every non-closed leaf is not proper and $\mathcal{F}$ is recurrent.

A foliation $\mathcal{F}$ is said to be R-closed if $\{(x,y)\in M\times M:y\in\overline{F(x)}\}$ is a closed subset.

2. A support function on a convex domain is the signed distance. Let $Q$ be a closed strictly convex domain around the origin. Then its support function is given by $h(\phi)=\sup\{x\cos\phi+y\sin\phi:(x,y)\in Q\}$. It is easy to see that the supreme is attained at a point on the boundary, whose oriented tangent line has angle $\phi$ with positive $y$-axis. Using this parameter the billiard system admit a coordinate $(\phi,\theta)$, where $\theta$ is the angle of the out-going vector with the tangent direction. In particular ${\bf v}(\phi,\theta)=e^{i(\theta+\phi+\pi/2)}=\langle -\sin(\theta+\phi),\cos(\theta+\phi) \rangle$.

Let $C$ be a closed piecewise-smooth convex curve around the origin, $\phi$ the angle of the tangent line at a point $(x,y)\in C$ with $y$-axis (serving as a parametrization, so $(x(\phi),y(\phi))$ and the line $L(\phi)$). Then the distance from $o$ to the tangent line $L(\phi)$ is $h(\phi)=x\cos\phi+y\sin\phi$.

Taking derivative with respect to $\phi$, we get $h'(\phi)=x'\cos\phi+y'\sin\phi-x\sin\phi+y\cos\phi$.

1. Phase transitions in statistical mechanics. A phase transition occurs when a material changes its properties in a dramatic way. For example water, as it is cooled and turns into ice. Phase transitions are characterized by an order quantity (like density) that changes as a function of a parameter of the system (such as the temperature). The special value of the parameter at which the system changes its phase is the system’s critical point.

A bifurcation occurs in a dynamical systems, when a small/smooth change of the parameter values (the bifurcation parameters) of a system causes a sudden ‘qualitative’ or topological change in its behavior. For example the ‘period-doubling bifurcation’ of Logistic map, the saddle-node bifurcation.

Phase transition in dynamical systems

– the parameters $t$‘s where the pressure $P(f,t\phi)$ fail to be $C^k$ for $k=0,1,\cdots,\infty,\omega$;

– the parameters $t$‘s where the system $f_t:M^2\to M^2$ shifts from integrability to nonintegrability, from regular to chaotic.