Chronology of the KAM Theorem and the Outer
Billiards Map 
●1892:
Poincare J.H. Les methodes nouvilles de la mechanique
celeste, Paris: GauthierVillars In 1887, in honor of the
60th birthday of King Oscar II of Sweden, a prize was offered to anyone who could
solve the following problem, due to Karl Weierstrauss: "Given a system of arbitrary mass points that
attract each other according to Newton’s laws, under the assumption that no
two points ever collide, try to find a representation of the coordinates of
each point as a series in a variable that is some known function of time and
for all of whose values the series converges uniformly." This was the famous
nbody problem which had its beginnings in the work of Isaac Newton
(16421727) for the Sun, Earth and Moon. But this 'lunar' problem was not trivial
and the general nbody solution, taking into account the gravitational
interaction of the planets, had proven to be very difficult. In 1710 Johann
Bernoulli solved the problem for 2 bodies but his methods did not generalize
to 3 or more bodies. At age 35, Henri Poincare submitted the prizewinning
paper which showed that there was no way to solve the problem by traditional
methods of reducing the dimension using 'integrals' of motion so there were
no integralbased 'closed form' solutions. By contrast the 2body problem is
'integrable'. The key issue was
resonances and 'small divisors' which could lead to unstable (homoclinic) orbits. (This left open the possibility that
the nbody problem could be solved by other means, and indeed the 3body was
solved by Karl Sundman in 1906 and the nbody
version was solved by Quidong Wang in 1991, but the
infinite series used in their solutions converged so slowly that they were of
no practical value. And this created a debate among mathematicians about the
definition of a 'solution') Poincare's paper was
later expanded to the three volumes of Les methodes nouvilles de la mechanique celeste and this laid the
foundation for new areas of study based on
qualitative methods. This spelled the end of the 'clockwork universe' of
Newton and Leibniz. Together with G. D. Birkoff,
Poincare laid the groundwork for topology and the study of manifolds. In his
(revised) paper for the prize, Poincare realized that even for 3 bodies there
could be solutions which had 'sensitive dependence on initial conditions' and
this would make prediction impossible. This condition is now the key
ingredient of chaos theory. Below is a chaotic orbit for the 3body problem. 
●1942: Siegel C.L. Iteration of (Complex) Analytic Functions,
Ann of Math 42: 607612 The
real issue for the nbody problem is whether there are 'stable' solutions.
Poincare was of the opinion that 'most' solutions would be unstable because
of the small divisor problem. The small divisor problem has its source in
irrationally indifferent fixed points.These are
'elliptic' fixed points which are neither attracting or repelling and where
the corresponding linear map is an irrational rotation θ. In complex
analysis these points are know as
holomorphic germs. Poincare stated the problem as follows: Given a
holomorphic (complex analytic) function such as f(z)
= ρz + z^{2}, with ρ = e^{i2}^{πθ}, for what values of θ will f
be linearizible near the fixed point z = 0 ? ^{ } For
such fixed points it is not clear whether the system will be stable under
small perturbations, because nearby rational rotations lead to small divisors
and are typically not stable. In 1942 Carl Siegel showed that for complex
analytic functions, the small divisor problem can be overcome for these fixed
points if θ satisfies a Diophantine condition which guarantees it cannot
be approximated closely by rationals. This showed
that, from the standpoint of complex analysis, the small divisor problem was
a matter of degree, so some solutions will converge and others will diverge. In
Paris, Gaston Julia (18931978) and Pierre Fatou
(18781929) were also interested
in this problem. They were trying to classify fixed points of complex
rational functions based on stability and the irrationally indifferent case
was the most difficult. Fatou thought that such
points were never stable because of the small divisor problem. On the left
below is the Julia set for an irrationally indifferent fixed point and on the
right is the Jilia set for a rationally indifferent
fixed point. In this case, the chaotic dynamics reach all the way to the
fixed point so there is no stable neighborhood, while on the left the fixed
point is surrounded by invariant rings which testify to the local stability.This stable region is known as a Siegel Disk.
The boundary of this disk is typically very complex. 
●1954: Kolmogorov
A.N.
Preservation of Conditionally Periodic Movements with Small Change in the
Hamiltonian Function, Doklady Academii Nauk (Report of the Academy of Sciences) SSSR vol.
98 : 527530. In the early 1950's A.N.Kolmogorov
was the most influential mathematician in Russia, but his work not widely
known to the outside world because of the Iron Curtain.
In an 1954 address to the International Congress of
Mathematicians he laid out a plan to resolve Poincare's stability problem
using new iterative convergence methods. Starting with an integrable
Hamiltonian system, the issue is whether the system can remain stable under
small perturbations. Jurgen Moser,
wrote a review of Kolmogorov's proposal and pointed out that there were still
some important unresolved issues. Moser felt that there was promise in
Kolmogorov's ideas and he set out to resolve these issues on his own. Even
though Moser was a friend and colleague of Carl Siegel, he did not base his
work on complex analytic functions because these were too 'well behaved' to
apply to a wide class of Hamiltonian (conservative) systems. He chose to
assume a 'smooth' realvalued function and meanwhile a brilliant young
protégé of Kolmororov named V.I. Arnold was
developing a proof based on real analytic Hamiltonian functions.

●1962: Moser
J.K. On invariant curves of areapreserving mappings of an annulus, Nachr. Akads. Wiss, Gottingen, Math. Phys. The main result is called the Twist
Theorem and it addresses the 'center problem' of Poincare, Siegel and
Kolmogorov. The critical case occurs when the center is an 'elliptical' fixed
point of an integrable Hamiltonian system. The
question is whether this fixed point can remain stable under periodic
perturbations. To simulate this, Moser used a mapping of the annulus of the
form T(ρ,φ) =
(ρ, φ + 2πω). For
a fixed radius ρ, the amount of 'twist', ω ,
is called the ‘winding number’. It is
assumed to be a smooth function of ρ. For fixed ρ, T(φ)= φ + 2πω so points would rotate
by 2πω and the orbits can be classified as rational or irrational.
The irrational rotations are called quasiperiodic and they are dense in the
circle as shown below. The rational rotations yield discrete orbits. Under periodic perturbations, the
rational orbits may break down and form rings of unstable fixed points and
stable 'islands' as shown in the middle. This break down is caused by
resonances between the orbit and the perturbation. Each of these new fixed
points feels the same perturbations so they behave in a similar fashion,
creating endless chains of fixed points and 'islands'. If enough of the
quasiperiodic orbits survive, this instability is localized. Moser showed
that that under certain 'smoothness' conditions on the Hamiltonian function,
a finite measure of these invariant quasiperiodic curves will survive the
perturbation and guarantee longterm stability. 
●1963:Arnold, V.I. Proof of a theorem by A. N .Kolmogorov on
the invariance of quasiperiodic motions under small perturbations of the
Hamiltonian.  Usp. Mat.
Nauk. SSSR 18, no. 5 Moser's Twist Theorem
and Arnold's proof form what is now known as the KAM theorem named after the
three authors. Both authors give credit to Kolmogorov for laying the
foundation for one of the most important results in stability theory. As
mentioned above, Arnold assumed a real analytic function and Moser assumed a
smooth function. Because of the
Iron Curtain, there was very limited exchange of information. In the end, the
Russians though Moser's approach was superior (because real analytic
functions are a proper subset of smooth functions), but Moser felt that
Kolmogorov's insights were the key element in the KAM Theorem. Moser also acknowledged the groundbreaking
work of Carl Siegel. (However if the four contributors were listed in
chronological order, the KAM theorem would be known as the SKAM Theorem.) In his 1972 text Stable
and Random Motions in Dynamical systems Moser looks at the KAM stability
issue from the standpoint of smooth functions as well as analytic functions.
He also discusses the small divisor problem in the context of complex
analytic functions and presents a simplified version of Siegel's proof. On the left below is a
rational torus of the type which can lead to resonances and stability
problems. This torus shows a 5:2 resonance between the two windings. This type of resonance exists between
Saturn and Jupiter. If ω_{J} and ω_{S} are the orbital frequencies of Jupite and Saturn, then 2ω_{J}  5ω_{S}
~ 0 and 'small divisor' terms of this type occur in the Fourier series
expansion of the motion. (Moser points out that Uranus and Neptune have a
ratio which is almost 2 and there is also an interesting relationship between
Jupiter, Mars and Earth: 3ω_{J}  8ω_{M} + 4ω_{E}
~ 0.) On the right is the
Standard Map which is often used to illustrate the breakdown of tori as
perturbations increase and the KAM theorem no longer applies.The
perturbation factor shown here is quite large. The magenta orbit is the
'last' invariant torus to survive because it has a winding number equal to
the Golden Mean and is therefore poorly approximated by rationals. 
●1971:C. L Siegel, J.K Moser . Lectures on Celestial Mechanics, SpringerVerlag, Berlin In 1952, Jurgen
Moser had just received a doctoral degree at Gottingen under Franz Rellich. He attended a lecture series on Celestial
Mechanics taught by Carl Siegel who had just returned from Princeton. Moser
was encouraged by Rellich to writeup the notes
from that lecture series. This was the beginning of a longterm collaboration
between Moser and Siegel. The lecture
notes were published in German in 1955 and translated into English in
1971 at which time Moser (now at the Courant Institute in New York) added
new material to the chapter on stability theory, to include existence
theorems for quasiperiodic solutions which formed the basis of the KAM
theorem.

●1973: J. Moser, Stable and Random Motions in
Dynamical systems, Ann. of Math. Studies, 77, Princeton Univ. Press, Princeton, NJ, MR0442980 (56:1355) This was based on a series of five Hermann Weyl lectures given at the Institute for Advanced Study at
Princeton in 1972. It includes extensive historical and mathematical
background behind the Twist Theorem and the KAM theorem. The Tangent Map
illustration below is from the Introduction. He states "As a consequence of Theorem 2.11 (the
Twist Theorem) below one can show that
every orbit is indeed bounded if is six times continuously differentiable (C^{6})"
Moser notes that the 'smalldivisor' issue arises even in this simple
scenario. The full proof of this boundedness was
given by Douady in 1982. In his original 1962
proof, Moser needed 333 continuous derivatives (C^{333}) but this was
later refined by Russman to a C^{5} curve
and Moser conjectured that it even may hold for C^{2}.
Counterexamples exist for the C^{1} case (F,Takens 1971). Circles and ellipses have no
limitations on derivatives so they are C^{∞}. In these cases
the orbits are circles or ellipses. For the general C^{6} case, the
orbits are bounded but may not lie on a smooth curve. The C^{2+epsilon}
conjecture is still open. Moser makes no mention of the polygonal case in
this work. 
●1978: J. K.
Moser Is the Solar System Stable ?
The Mathematical Intelligencer, Vol. 1, No. 2: 6571 This was the first 'popularpress' publication about
the KAM Theorem and it appeared in the inaugural issue of the Mathematical Intelligencer which is
devoted to mathematics and its historical development. It started as a
pamphlet in 1971. The cover illustration
is the 'smooth curve' version, but Moser also presents the polygonal
version as an example where the KAM theory breaks down. He attributes the OuterBilliards
Map 7 to B.H. Neumann in 1959 but also mentions P.C.
Hammer's 1963 work in convexity. Over the next 20 years S. Aubry and J.
Mather developed a theory to describe what happens when the invariant KAM
curves break down. This is now known as AubryMather Theory and it is
concerned primarily with monotonic twist maps and the corresponding closed
invariant Cantor sets. (The set of KAM curves has finite measure but they
form a Cantor set with no interior so it is impossible to pick an initial
condition and say whether it is stable or not.)

●1987 F. Vivaldi, A. Shaidenko, Global stability
of a class of discontinuous dual billiards. Comm. Math. Phys. 110 , 625–640.
MR895220 (89c:58067) The authors show that regular polygons
and lattice polygons must have bounded orbits under 7 because there are endless rings of 'necklaces' such as the one shown
below. The region between the rings is invariant, so orbits are trapped.
These rings mimic the quasiperiodic orbits of the continuous case. For
regular ngons, the necklaces are rings of 2ngon 'Dads' as shown on the
right for the regular haptagon. In the case of lattice polygons, the fact
that orbits are bounded implies that they are periodic. This is not true for
polygons in general and in fact the regular pentagon has orbits which are
bounded but not periodic. 
●1988
Chua L.O.,Lin T. Chaos in digital
filters. IEEE Transactions on Circuits and Systems 35:648658 In 1968, J.B. Jackson and his colleagues at Bell
Labs were trying to understand the source of selfsustaining oscillations in
digital filters. Scientists were just becoming aware of the possibility of
chaotic behavior in feedback networks. One of the pioneers in electrical
circuit chaos was L.O. Chua at Berkley. He showed that certain digital filter
circuits had the potential for chaotic behavior. Some of the discrete
mappings used to describe these filters show a remarkable ability to 'shadow'
the outerbilliards map when the polygon in question is regular. This
connection is discussed in Outer Billiards, Digital Filters and Kicked
Hamiltonians at http://arxiv.org/abs/1206.5223. 
●1995 Tabachnikov, S. On the dual billiard problem.
Adv. Math. 115 (1995), no. 2, 221–249. MR1354670 One
of the pioneers in the study of outer billiards (and inner billiards) is
Serge Tabachnikov. In 1995 he showed that the regular pentagon has a
nonperiodic orbit which is dense in the inner star region. The web for this
region can be generated using the orbit of this point. The web is fractal
with dimension Ln[6]/Ln[1/GenScale[5]] ≈ 1.241. The first 50,000 points in this
nonperiodic orbit are shown below.
The pentagon N = 5 is also known as M. It is shown here in 'standard
position' with vertex 1 at {0,1}. The coordinates of
a nonperiodic point with dense orbit are: s = {M[5]][[1]],
M[4]][[2]]}  where M[[5]] [[1]] is the first coordinate of vertex 5 of M.
The magenta arrow extends from M[[5]] to s. Since
s is on a forward extended edge of M, it cannot have a periodic orbit because
it has no inverse image. So points on a forward edge either map to a trailing
edge or they are nonperiodic. Any point that maps to a trailing edge must
have been a point of discontinuity of the outerbilliards map. All
discontinuities of 7 for a regular polygon can be uncovered by mapping
the forward edges. This is called the (forward) web and it is the closure of
the plot below. Mapping the trailing edges under 7^{1 }will yield the
same web in the limit. This is called the inverse web. These webs are only
fractal for N = 5, 8, 10 and 12. Any
twiceodd pair such as N = 10 and N = 5 will have congruent webs. It is no
coincidence that all of these Ngons have Phi[N] = 2
where Phi is the Euler totient function – so they
are classified as ‘quadratic’ polygons. 
●1995
Davies A.C. Nonlinear oscillations and
chaos from digital filter overflow. Philosophical Transactions of the
Royal Society of London Series A Mathematical Physical and Engineering
Sciences 353:8599 A.C. Davies and others used the mathematical
models of digital filter circuits devised by Chua & Lin in 1988 to
develop a dynamical theory based on 'mod2' mappings of the unit square to itself. These 'torus' mappings (shown in magenta below)
yield elliptical centers that mimic the perturbed fixed points of a
Hamiltonian system. The elliptical centers can be rectified to circles as
shown in the blue rhombus. For certain parameters, the rhombus is an exact
copy of the local web for the outerbilliards map. This provides unique
approach toward analyzing the dynamics of regular 2ngons. When n is odd, the
web of the ngon is a subset of the 2ngon web. Below is the case of N = 14. 
●1997:Ashwin
P, Elliptical behavior in the sawtooth
standard map, Physics Letters A 232:409416 The ChirikovTaylor Standard Map was devised in
the early 1970's. It has become a paradigm for Hamiltonian chaos
in the sense that it is an area preserving map with divided phase space
where 'integrable' islands of
stability are surround by a chaotic component. There have been a number of
'nonsmooth' versions of the Standard Map. P.Ashwin devised a version using a
sawtooth nonlinearity which he showed is equivalent to the digital filter
map used by A.C. Davis. Below is the sawtooth standard map in magenta and the
equivalent digital filter map in blue. This provides a connection between the
Standard Map and the outerbilliards map. See http://arxiv.org/abs/1206.5223. 
●2007: R. Schwartz, Unbounded orbits for outer
billiards. I. J. Mod. Dyn. 1 , 371–424.MR2318496 (2008f:37082) When J. Moser discussed the polygonal
version of the outer billiards map in 1978, he used it as a model of what
might happen when the KAM theorem broke down, so the major issue was
"Are there unbounded orbits?". The became known as the MoserNeumann question. In a sense it
is the discrete analog to the question of the stability of the solar system 
because these are the issues addressed by the KAM Theorem. In 2007, R.E.
Schwartz discovered a class of unbounded orbits for a convex quadrilateral
known as a Penrose Kite. In his more recent 2009 monograph Outer billiards on kites , Richard
Schwartz shows that the Tangent Map has unbounded orbits on any irrational
kite  that is any kite with a single irrational vertex. He conjectures that
this is 'generic' for large classes of polygons. On the left below is a point
p which was shown to be nonperiodic. The numbers given are the periods of
some major 'resonances'. The orbit of p lies entirely on a 1dimensional
lattice composed of lines of the form y = k where k is an odd integer. The
illustration on the right is a colored version of this same web, used in the Math Book by Clifford Pickover. 
●2008: . D. Genin, Research announcement:
boundedness of orbits for trapezoidal outer billiards. Electronic Research Announc. Math. Sci. 15 , 71–78.
MR2457051 (2009k:37036) The mathematical definition of a trapezoid is a
quadrilateral with at least two sides parallel, so the parallelograms are
special cases of trapezoids. There is no affine transformation that would
turn a trapezoid into a kite. When a trapezoid is a lattice, it is known that
all orbits are bounded, but D. Genin proved that this is true for all
trapezoids. The web for an arbitrary trapezoid is shown below. 
●2009. D. Dolgopyat, B. Fayad, Unbounded orbits
for semicircular outer billiard. Ann. Henri Poincare 10 357–375. MR2511890 (2010d:37076) The MoserNeumenn question was directed at both
smooth (differentiable) curves and polygons. For smooth curves, the only
known integrable cases are the circle and the ellipse. In both cases the
Tangent Map preserves circle and ellipses, so these are integrals of motion.
This is true for inner billiards as well. For inner billiards, George
Birkhoff conjectured that these are the only two integrable cases. This has
not been proven, but it may hold for both inner and outer billiards. Moser and Douady used the Twist Theorem to show that
if the generating curve is C^{6} (six times continuously
differentiable), then all orbits are bounded.A semicircle has elements of
both the smooth and nonsmooth curves. Like a polygon, it is C^{0}
(continuous) but not C^{1} (differentiable). In 2009 D. Dolgopyat and B. Fayad showed that a
semi circle has unbounded orbits that diverge in spiral patterns. Below is an
approximation to a semi circle using a truncated regular polygon with 200
sides. The magenta orbit on the right, started in close proximity to the
semicircle  which is just a speck on this scale.This orbit shows typical
divergent behavior. 
●2010 R. Schwartz, Outer Billiards, Arithmetic
Graphs, and the Octagon. arXiv:1006.2782. The
cases of N = 5, 8 , 10 and 12 are the only known
cases where the web structure is a perfect fractal. In all these cases it
appears that a single nonperiodic orbit is dense in the web but this has
not been proven for N = 12. As described above the fractal dimension of the
regular pentagon is Ln[5+1]/Ln[1/GenScale[5]]
≈ 1.241 and it is not hard to
show that the fractal dimension of the regular octagon is
Ln[8+1]/Ln[1/GenScale[8]] ≈ 1.246. There
is only one nonredundant way to 'remap' the vertices of a regular octagon:
{1,2,3,4,5,6,7,8} →{1,4,7,2,5,8,3,6}.This mod3 remapping of the
nonperiodic orbit is what generates the fractal Koch snowflake on the right
below. The mod2 remapping is redundant but it generates the fractal 'sponge'
in the middle, and the mod1 'remapping ' is the original (return) orbit
which is shown on the left below.
These are called 'projections' since they map R^{8}→R^{2}. They are also examples of the
arithmetic graphs that Richard Schwartz used to prove that the Penrose kite
had unbounded orbits. 
●2012 G.H. Hughes, Outer Billiards on Regular Polygons  arxiv.org/abs/1311.6763 In 1989 F.Vivaldi and A. Shaidanko showed
that all outerbilliards obits of a regular polygon are bounded and in that
same year the author met with J. Moser at Stanford University to discuss the
‘canonical’ structures that always arise in the singularity set W when N is
regular. Moser suggested that a study of these structures would be an
interesting exercise in ‘recreational’ mathematics. This is exactly what it
became over the years  but the evolution of these canonical ‘First Families’
proved to be a difficult issue except for regular Ngons with linear or
quadratic complexity. This paper introduces these canonical First Families
and their connection with the outerbilliards map. Below is the First
Family of N = 14. Note that the odd S[k] and DS[k] have two distinct forms as
magenta 14gons and cyan 7gons. Such S[k] are called ‘androgynous’ because even and odd Ngons have
different dynamics and the hence different ‘parity’ or ‘gender’. Only
twiceodd Ngons can support both genders in their First Families. By the
Twiceodd Lemma, there is an invertible transformation T that would map this
First Family of N = 14 to the First Family for N = 7. In this transformation,
T[S[5]] would be N = 7 so the cyan S[5] here can be
regarded as a scaled copy of N = 7. This is true because N = 14 and N = 7
have equivalent cyclotomic fields and all the algebraic structure of a
regular Ngon is contained in the cyclotomic field Q_{N}. As a vector
space over the rationals Q, the cyclotomic field is
generated by the (complex) vertices of N. The sequel article below has
recently been updated. It shows First Family scales can be used to generate
the maximal real subfield of Q_{N} which is denoted Q_{N}^{+}. This is called the scaling field of N. 
●2019 G.H. Hughes, Families of Regular Polygons and their Mutations  arxiv.org/abs/1612.09295 As illstrated
above, every regular Ngon generates a canonical ‘family’ of regular polygons
which are conforming to the bounds of the ‘star polygons’ determined by N. These
star polygons are formed from truncated extended edges of the Ngon and the
intersection points (‘star points’ shown above in blue) determine a scaling
which defines the parameters of the family. Based on a 1949 result of Carl
Siegel it follows that this starpoint scaling forms a unit basis for the
maximal real subfield of the cyclotomic field Q_{N}. The traditional
generator for this subfield is 2cos(2π/N) so it
has order Phi(N)/2 where Phi is the Euler totient
function. This order is known as the ‘algebraic complexity’ of N. The family
of conforming regular polygons shares the same scaling and complexity as N,
so it is called the First Family of N. This scenario is
illustrated in steps (i) and (ii) here for N = 14. (The cyan First Family
shown here is identical to the illustration above.) This is independent of
any mapping but for piecewise ‘rational’ isometries such as the
outerbilliards map tau, the
digital filter map Df , or the dualcenter
map Dc, the singularity set W can
be formed by iterating these extended edges of N under tau , Df or Dc. This is
illustrated in (iii) below. This singularity set W is generated by the
outerbilliards map tau but Df and Dc will generate congruent versions of W. This is also called a ‘generalized
star polygon’ since it is generated by the edges of the star polygon of N. It
retains the same dihedral symmetry group as N.
Section
4 shows that W preserves the S[k] and Lemma 4.1 says that tauorbit of each S[k] has a constant stepk periodic orbit. Therefore the
S[k] mimic the classical periodbased ‘resonances’ from
Hamiltonian dynamics and the Standard Map. (See the illustration below). In
addition the ‘web’ W can be regarded as the disjoint union (coproduct) of the local webs of the S[k]. These local
webs are still very complex but for Ngons with complexity that is linear or
quadratic (N = 3,4,5,6,8,10,12), this S[k] evolution
is sufficient to describe the resulting topology of W  because in these
cases there is never more than one effective scale. Therefore the topology is
linear for N = 3, 4 and 6 and simple fractal for N = 5,8,10,12.
When there are multiple
nontrivial scaling parameters, the topology of W may be multifractal and
there is no guarantee that the First Family scaling will be sufficient to
determine the geometric scaling of W  but the Scaling Conjecture says that
this scaling in Q_{N}^{+ }is always sufficient. Below is a connection
between the Standard Map – which simulates the phasespace dynamics of the
KAM Theorem  and the digital filter map Df
– which generates a web that is conjugate to W. At the bottom we show the connection with
the complexvalued dualcenter map Dc.
This map also generates a web which is conjugate to W. These connections are
described in Outer
Billiards, Digital Filters and Kicked Hamiltonians. This last image shows a remarkable chain of
equivalences that begins with the Standard Map – because P. Ashwin devised a ‘sawtooth’ version
of the Standard map which allows it to be ‘tuned’ to N = 14 (shown here in magenta) . Under a simple
change of variables this sawtooth map is equivalent
to the digitalfilter map (in blue above) and with the right parameters the
digitalfilter map is conjugate to the outerbilliards web W for N = 14 (in
black). Since the Standard Map is just a graphical
representation of the phase space for the KAM Theorem, this chain shows a
direct connection between the ‘resonant’ islands of stability of the KAM
Theory and the S[k] of the First Family. 

* Thanks to John Mather of
Princeton University for his insights into the mathematics and politics of
the KAM Theorem. 