Let be a homeomorphism on a compact space . Given a ceiling function , we consider the mapping torus and the suspension flow on , which is just the flow on , respecting the equivalence relation .

Let be an exact symplectic manifold, be a Hamiltonian diffeomorphism. Then for any primitive 1-form , that is, , consider the 1-form on . When does it descend to a 1-form on ? If so, then is a contact form on and is the corresponding Reeb flow on .

Recall that for some function on . Therefore, descends to a 1-form on if and only if is constant. Note that different choices of result in an additional coboundary of and hence an additional coboundary of the ceiling function on .

A serious issue about the assumption of exactness of is that closed symplectic manifolds are never exact by Stokes’s Theorem. So in order to construct a contact flow using a symplectic map on a closed symplectic manifold , one can blow up the manifold at an elliptic fixed point or along an elliptic periodic orbit, and denote the new manifold by . Then we extend to on , construct the suspension *.* It is a contact manifold with boundary. One can glue a solid torus twisted along the boundary component of and obtain a closed contact manifold.

Let be an ellipse with , be the billiard map on the phase space . Note that is a monotone twist map. The rotation interval of is . Moreover, for any , , there is a unique invariant curve of rotation number . The case is special: it consists of a periodic orbit of period and two pairs of heterolinic connections between them, say the upper pair and the lower pair . These pairs are invariant but not smooth: both are singular at the periodic orbits. However, they are the only two ways to form invariant curves of rotation number : is smooth, but not invariant.

Now let , be a covering space of that is copies in the horizontal direction and then copies in the vertical direction. It can be viewed as a standard annulus. Let be the lift of to the new annulus . Then the rotation interval is . For each , , there are exactly two invariant curves of rotation number , both having singular points.

Let be the stand sphere with the induced Riemannian metric, be the unit tangent bundle of . There is a natural map from to . That is, let , set . It is easy to see that . Moreover, is bijective and hence a homeomorphism.

Let , be the equator, which is a simple closed geodesic. Let be the lifted closed orbit of the geodesic flow on , and be the corresponding curve of matrices. Note that .

Let , , be a family of closed curves that deform to the trivial curve at the north pole . That is, , where . For each , let be the curve induced by . Let , . This process is not defined directly for . But the limit does exist. Therefore, we denote the limit as . It is easy to see that , which coincides with .

Similarly, one can deform to the unit tangent bundle at the south pole say . Note that both cycles wrap around the -axis counterclockwise. However, the two normal directions (aka the orientation) at and are opposite. So we have . It follows that , while . More generally, one can show that a smooth cycle with transverse self-intersections is contractible in if and only if it has an odd number of self-intersections. In particular, . This is a topological invariant. Hence it holds for all metrics on .

Consider a monic polynomial with integer coefficients: , .

The complex roots of such polynomials are called algebraic integers. For example, integers and the roots of integers are algebraic integers. Note that the Galois conjugates of an algebraic integer are also algebraic integers.

Consider a square matrix with positive integer entries. By Perron-Frobenius Theorem, there is a unique positive eigenvalue . Moreover, it admits an eigenvector of all positive entries, and satisfies for any other eigenvalue of . In particular, for any of its Galois conjugates. Such a number is called a Perron number. More generally, a weak Perron number is a real algebraic integer whose modulus is greater than or equal to that of all of its Galois conjugates.

Let be a continuous map on the interval , and the topological entropy. Assume is postcritically finite: is a finite set. Then the partition of along the postcritical set is a Markov partition for , since (1) the endpoints are sent to endpoints, (2) every folding corresponds to a critical point of . Therefore, is the leading eigenvalue of the incidence matrix associated to the Markov partition . It follows that is a weak Perron number.

Let be a group and be a topology on . Then is said to be a topological group if the two structures on are compatible: both and are continuous.

(1) any topological group can be made Hausdorff by taking an appropriate canonical quotient.

(2) any group with the discrete topology is a topological group: it is just an ordinary group.

We will assume all group topologies are Hausdorff and non-discrete. More generally, a group is topologizable if it admits a non-discrete Hausdorff topology. The following is based on the article.

[Markov 1944] A subset is unconditionally closed (unc-closed) if it is closed in every Hausdorff group topology on . For example, if is unc-closed, then for any group topology on , is an open set. Then the topology must be discrete. So is unc-closed if and only if is non-topologizable.

The collection of unc-closed sets form a topology on , which is called the Markov topology . Note that is , but not necessarily Hausdorff. We can reformulate the above observation as: is non-topologizable if and only if is the discrete topology.

[Markov] A subset is elementary algebraic if there exist , and such that . Then a subset is algebraic if it can be written as an (arbitrary) intersection of finite unions of elementary algebraic subsets of . This resembles the property of being closed sets. [1976, 1999] The Zariski topology is the topology in which closed sets are precisely the algebraic subsets of . For example is an ele-alg subset. So the center is an algebraic subset. It is easy to see that algebraic sets are unc-closed. Therefore, is coarser than .

A natural question is: when does ? Markov proved it for countable groups, and Perel’man proved it for abelian groups. There are groups for which .