Chronology of the Outer Billiards Map (Tangent 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
Tangent Map 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. Being close to the fixed point does not guarantee
stability.) 
●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 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 Tangent Map when the polygon in question is regular. 
●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. It 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 M is shown in 'standard
position' with vertex 1 at {0,1}. The coordinates of a nonperiodic point
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. If a point maps to a trailing edge, it was a point
of discontinuity of the Tangen Map.All discontinuities of the Tangent Map 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 the inverse Tangent Map will yield the same web in the
limit. This is called the inverse web.

●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 mathematical models of
digital filter circuits 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 tangent map. This provides a 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 Tangent Map. 
●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 regular
pentagon and regular octagon are the only known cases where the web structure
is a perfect fractal. In both cases a single nonperiodic orbit is dense in
the web. The fractal dimension of the regular pentagon is
Ln[5+1]/Ln[1/GenScale[5]] ≈
1.241 and 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. 
* Thanks to John Mather of Princeton University
for his insights into the mathematics and politics of the KAM Theorem. 