Example. (1) When is a rotation, then is stable. (2) When is a hyperbolic fixed point, then it is unstable. (3) Using complex coordinate , let . Then . Picking and letting , we see that is not stable; picking and letting , we see that is not unstable. Therefore, is mixed.

Example. Let . Consider the rotation and its nonlinear perturbation . Then whenever . It follows that is unstable.

]]>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.

Now suppose is a topological space. There is a natural -algebra, the Borel -algebra on . Then a measure is called Borel if .

Regularity. A Borel measure is inner regular if for any open subset , over compact subsets . It is outer regular if for any Borel subset , over open subsets . It is locally finite if for any , is finite for some neighborhood . Note that if is locally compact, then being locally finite is equivalent to that is finite for any compact subset .

There are different ways to define Radon measures.

Definition 1. A Borel measure is called a Radon measure if it is inner regular, outer regular and locally finite.

Definition 2. A Radon measure is a (positive) continuous linear functional on , where is the vector space of real-valued continuous functions with compact support.

By Riesz-Markov representation theorem, the two definitions are equivalent.

]]>Let be the set of points such that . It follows that we can reinterpret the function as . In fact, there exists a function such that and . Such a function is called a generating function of .

For the existence of a generating function, a necessary and sufficient condition is . The last equation holds since preserves the 2-form .

Then we compare the two expressions and and get a formula for the generating function.

In the other direction, if such a generating function exists, then preserves the 2-form .

]]>After Newton, one might wonder why the nature behaves this way. When we throw a stone in the air, the stone doesn’t know any math and still ‘knows’ where to land. This leads to the least action principle in Lagrangian mechanics: among all the possible choices of paths from one point to another (in the configuration space), the one with least action is the physical one. This leads to the Euler-Lagrange equation.

The Euler-Lagrange equation is a second order different equation about the motion . One can introduce the velocity variable and reduces E-L to a system of first order differential equations on the phase space . However, the different equation about is implicit.

The Hamiltonian mechanics is kind of dual to the Lagrangian mechanics. It involves the momentum instead of the velocity . This is not an artificial change: the new space admits a natural (symplectic) structure, and the differential equations of motion become explicit: it is given by the Hamiltonian vector field of the Hamiltonian function.

]]>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 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 .

]]>More generally, one consider a time-dependent vector field. That is, , where may change in time. This in turn genertes a one-parameter family (non-autonomous) of maps , , such that .

]]>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.

]]>Recall that for a general symplectic manifold , the one forms are in 1-1 correspondence to vector fields via . Then being exact is equivalent to the existence of a Liouville vector field — a vector field satisfying .

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