Thanks also go to Penn State University for financial support in the years 1990-1992. The Aubry–Mather theory proved the existence of invariant circles and invariant Cantor sets (the ghost circles) for the area-preserving, monotone rotation maps of annulus or of cylinders. Bernstein-Katok found Birkhoff periodic trajectories, seen as traces of missing tori, for the system in the KAM theorem, but under the stronger condition that the Hamiltonian function is convex.
We find the "isolatirig block", a structure invented by Conley and Zehnder, to demonstrate the existence of the Birkhoff periodic circuit for the KAM system.
Poincare's Last Geometric Theorem and Periodic Orbits of
Note the small perturbation Tf of the area-preserving mapping T and consider two T-invariant circles+ and r- with rotation numbers A+. We will focus on three closely related aspects: periodic orbits of ring mappings, the KAM theorem, and Aubry-Mather theory. We begin by looking at the evolution at fixed points (period one period points) of ring mappings.
There are many research papers in this area regarding the generalization of Poincare's last geometric theorem.
KAM Theorem and Aubry Mather Theory
When the time interval becomes infinite, the effect of small perturbation cannot be ignored. It is a generalization of the monotone torsion condition on annulus maps and is equivalent to the non-degeneracy condition for the Hamiltonian function in the KAM theorem. Any region preserving the monotone rotation homeomorphism of the ring has quasi-periodic orbits of all frequencies belonging to the inten-a.l rotation.
The phase space-cylinder- is fused by the tori invariant y =constant. we get the small perturbations of the integrable system.
Aubry-Mather Set as Limit of Periodic Orbits
Finally, we note that although the Aubry-Mather theory established the existence of 'many' invariant Cantori, which are the counterpart of the invariant tori in the KAM theorem, the meaning of 'many' here is referred to the fact that for every admissible rotation number, there is at least a corresponding invariant set. The proof of the existence of the invariant set in Mather's theorem is thus reduced to the proof of the existence of Birkhoff periodic orbits of type (p, q) for any rational number belonging to the twist interval [ao, at] belongs. Based on this scheme, M.Muldoon [Mull] performed an ambitious calculation of the numerical approximation of Birkhoff's periodic orbits, looking for the Cantori for some four-dimensional systems, and got some interesting pictures.
It is interesting to see that, for all these three types of small perturbations, Muldoon's numerical investigations indicate the existence of the Cantori in these four-dimensional systems.
An Outline of Bernstein-Katok's Result
To find the periodic orbit with rotation vector ~, the equivalent is to find the critical point of Lw.~,q, i.e. it also suggests certain regular behavior of the Cantori, which has been an interesting problem in this field for a long time. One thing I want to emphasize: for all three types of small perturbations, the unperturbed systems have convex generating functions and thus fall into the class that Bernstein-Katok studied.
The main results are as follows: for a perturbed Hamiltonian system with n degrees of freedom, if the Hamiltonian function is convex (KAM's theorem only requires that the Hamiltonian function be non-degenerate), then there exist n distinct periodic orbits with arbitrary admissible rational rotation vector near the corresponding torus of the unperturbed system. It can be seen from (4) that rational vectors mostly do not satisfy the Diophantine condition. In other words, undisturbed toruses with rational rotation vectors are, in a sense, the easiest to break under perturbation.
Inspired by the Aubry-Mather theory, one can ask the question: where does the missing tori go? The image of the last geometric theorem was born when Poincare studied the behavior of the perturbed system near an undisturbed rational circle. Is there any periodic orbit with the same rotation number as the undisturbed circle.
In this case, flr=ro is a periodic map, where period q is equal to the least common multiple of all denominators of the components of a(r0. The regularity of the invariant circle and of the Aubry-Mather set is always an important factor. issue in the study.
The Regularity of Aubry-Mather Sets
If there is a periodic point with period q, then the elevation (x,r) of the periodic point satisfies Fq(x,r) = (x +w,r) for some wE zn. The earliest study of the regularity of invariant closed curves is by Birkhoff [Bir2]. A consequence of Birkhoff's theorem is that, if X is an invariant, homotopically nontrivial circle in S1 x R1, then X is the graph of a Lipschitz function.
Using Birkhoff's theorem, we obtain two wonderful results on the nonexistence of an invariant circle for standard mappings [Mat3] and the nonexistence of a caustic for planar convex billiards [Mat4]. And this regularity theorem also plays a key role in the Aubry-Mather approximation scheme determined by Birkhoff periodic orbits. Since the Aubry-Mather set is on the one hand the "trace" of the invariant circle, and on the other hand is part of the graph of the mapping from S1 to R1, we naturally hope that we can choose the mapping to be Lipschitz, in other words, the ghost circle has the Lipschitz property.
Numerical results by M. Muldoon seem to show that there are certain regularities between the approximate periodic Birkhoff orbits. All these results require the convexity condition of the generating function or the Lagrangian function. In the case where the generating function is non-convex, we expect the orbits to be very irregular for the high-dimensional system, as Herman's example (see Section 6) shows that perturbed periodic orbits can stray far from the unperturbed torus.
We use the standard variation method, as Bernstein-Katok did, to obtain the periodic orbits.
Variational Method for Finding Periodic Orbits
Conley-Zehnder's well-known paper (CZl,Conl] points out that for a function to have critical points, it is the topological type of the space that matters, whether the space is compact or not is not crucial. Crucially, they discovered is that the neighborhood of the minimal orbit has the necessary topology that guarantees the existence of n + 1 critical points. We find two crucial topological structures hiding behind the space XWJ,q· One is the topological type of the undisturbed torus and another is "insulation block".
Basically, these two are all that are needed to guarantee the existence of n + 1 critical points of LWJ,q. Furthermore, by using the size of the "insulation block", we are able to measure the distance between the perturbed Birkhoff periodic orbits and the unperturbed torus. Let me explain what the generating function is in our situation and why (R) is the non-degeneracy condition for the Hessian of the generating function. Dh(a(r)), we obtain that Dh = -a-1• Hence we conclude that the regularity of the mapping a is equivalent to h having non-degenerate Hessian.
We will consider the standard deviation as M. Muldoon does, viz. we will choose h of the form:. is a very simple symmetric matrix. The vector w I q is called the rotation vector of the point. gt;, r ), depending on the choice of lift F, but is uniquely defined modulo zn. We have thus reduced our problem to finding the critical point x E W"',q', which suffices xi+1 -xi E a(U), of the action LWJ,q·.
The general case proceeds along the same lines, and we will state the crucial part of the proof for the general situation after the proof for this simple case. In particular, the equilibrium configurations and periodic solutions of general elliptic type are stable when the number of degrees of freedom exceeds 2. The example happens to be a canonical mapping of four-dimensional space (Arnold calls the symplectic map a canonical map).
This estimate verifies the dependence of the distance on the period q and shows that our estimate (22) is optimal.
Minimal Geodesics and Lyapunov Exponents
I followed the idea to a theorem of E.Hopf and showed the following stiffness result: the integration of the Gaussian curvature, that is, of the average Gaussian curvature, along any closed minimal geodesic is always non-positive. Furthermore, by combining Katok's results, the variational entropy principle and Ruelle's inequality, it can be shown that most closed minimal geodesics. We believe that most closed minimal geodesics belong to case III, since negative curvature is the main cause of hyperbolism [Anol].
In the absence of a focus along a single minimal closed geodesic, the negativity of the mean Gaussian curvature implies the existence of the nonzero Lyapunov exponents. Let 11,12 be two infinite curves in D, they would be of the same type as. For every hyperbolic line lo in (D, u0 ) there exists at least one minimal geodesic 1 that is of type lo; on the other hand, for every minimal geodesic 1, 1 of the same type is a hyperbolic line lo in (D, a0.
For every periodic hyperbolic line lo in (D, uo) there is at least one periodic minimum geodesic 1 of type IO·. Because of the above facts and also because of a result about the word growth rate of the fundamental group by J.Milnor [Mill], we know that there are many minimal closed geodesics. For a geodesic 1 in M we speak of the Lyapunov exponents of 1 as the Lyapunov exponents of v, where v is a unit tangent vector to I·.
Our other results involve estimating the exponential growth rate of the number of surveyors from classes I, II, III. We can now give another description of the largest Lyapunov exponent x+( v ), for v such that 7r
J.Mather, Exi&tence of qua&i-periodic orbits for two&t homeomorph&m& of the annulus, Topology 21 no.
