8. **Definition**. Given a family of maps with corresponding invariant densities . Then is said to be *acim-stable* if implies .

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 be the 3D Heisenberg group, with . Let be a cocompact discrete subgroup (for example ). Then is a 3D nilmanifold. A general non-toral

three-dimensional nilmanifold is also of this form. Suppose we have a homomorphism , which is of the form , which induces a 2D-foliation, say on and on .

**Theorem. **Every Reebless foliation on is almost aligned with some .

Plante for , Hammerlindl and Potrie for .

Theorem. Every partially hyperbolic system on is accessible.

J. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures (convervative), Hammerlindl and Potrie (general)

6. Let and be a function. Consider the solutions of the recurrence relation:

() for all .

Note that () is actually a finite sum of terms over . It is the derivative of formal series with respect to .

**Example.** Billiards, or generally twist maps, where and is the generating function, the solution gives the configuration of an orbit.

There are some conditions:

**Periodicity.** . So descends to a map on .

**Monotone.** for all and all , and for all .

**Coercivity.** is bounded from below and there exists such that as .

Under these conditions the () is called a monotone variational recurrence relation.

A sequence is said to be a **global minimizer**, if (understand as over all intervals) for all sequences . Clearly a global minimizer solves (). The collection of global minimizers is also closed under coordinately convergence.

For a real number , a sequence with is called an -minimizer, if it is minimizes among all ‘s with .

Ana-minimizers in general need not be solutions to ().

Given a rational , we consider the operator (shift and subtract ) and Birkhoff orbits of rotational number prime sum over the periodic ones .

**Periodic Peierls barrier.** Let be a real and be coprime. Then as

.

It is easy to see that

There exists a periodic minimizer with if and only if .

gives an invariant curve if and only if .

Then the Peierls barrier at a general frequency is defined as when the limit exists (see Mramor and Rink, arxiv:1308.3073).

5. Let be a monotone twist map, be the generating function of : iff and . Then the twist condition is equivalent to .

Let's further suppose . A sequence corresponding to the orbit is called the configuration. In particular we have

() .

Consider a -family of the configurations. Then taking derivative of with respect to (), we get

() .

**Definition.** A sequence is called a **Jacobi field** along if it satisfies the equation

Then is said to **have conjugacy**, if there exists a nonzero Jacobian field along vanishing at some .

The map 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 is conjugacy-free. Then any two configurations and cross at most once.

*Proof.* Let’s assume they already cross once at : and Pick two linear functions connecting and , . Then we can solve from () connecting and for all other . Clearly and . If they cross again after , we see that , and the positive definition would force , contradiction.

4. Let be a metric space and be a connected open subset, whose -kernel is defined as . Clearly as . But may not be connected, since there might some narrow channels connecting . Let be small enough such that . Then for all , let be the connect component of containing the marked point .

**Proposition.** .

*Proof.* It suffices to show that is closed in . So pick a point . Let such that , and such that . Then we actually have . This ends the proof.

3. Let be a closed manifold (mostly 3 dimension) and a foliation on . A leaf is closed if (like a periodic orbit or a genus- surface). It is proper if is closed (like the graph of ). It is recurrent if it is either closed (trivial proper) or non-proper (like a line on with irrational slope). Let be the part of closed leaves (note that may not be closed, like the periodic orbits of Anosov flow). Let be the part of proper leaves (nontrivial proper: not closed) . In particular all leaves outside are recurrent.

The foliation is said to be non-wandering if has no interior. And is said to be recurrent if , that is, every leaf of is recurrent. Moreover, is said to be almost periodic, if forms a new decomposition of (either disjoint or coincide). Yokoyama observed the following proposition:

Proposition. A almost periodic foliation is recurrent.

Proof. Let be a non-closed element and . Clearly . Then almost periodicity implies that . So . So every non-closed leaf is not proper and is recurrent.

A foliation is said to be R-closed if is a closed subset.

2. A support function on a convex domain is the signed distance. Let be a closed strictly convex domain around the origin. Then its support function is given by . It is easy to see that the supreme is attained at a point on the boundary, whose oriented tangent line has angle with positive -axis. Using this parameter the billiard system admit a coordinate , where is the angle of the out-going vector with the tangent direction. In particular .

Let be a closed piecewise-smooth convex curve around the origin, $\phi$ the angle of the tangent line at a point with -axis (serving as a parametrization, so and the line ). Then the distance from to the tangent line is .

Taking derivative with respect to , we get .

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 ‘s where the pressure fail to be for ;

– the parameters ‘s where the system shifts from integrability to nonintegrability, from regular to chaotic.