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Genin
ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
Research Announcement:
boundedness of orbits for
trapezoidal outer billiards
Daniel Genin, National Institute of Standards, Washington DC, USA
Article originally published
in “Electronic Research
Announcements in
Mathematical Sciences”,
V. 15 (2008)
http://www.math.psu.edu/era/
Reprinted with permission
ABSTRACT. We state and briefly sketch the proof of a result
on boundedness of outer billiard orbits for trapezoidal outer
billiards.
KEYWORDS: Bounded orbits, Geometric concepts, KAM Theorem, Outer billiards,
Periodic orbits
Introduction
Outer billiards were introduced by B.H. Neumann in about 1960
(Neumann, 1959) and later popularized by J. Moser in (Moser,
1973). Moser used outer billiards as an example of a problem
amenable to the methods of KAM. In this article it was shown
that for outer billiards with a six times differentiable boundary
all orbits must be bounded. Moser then went on to ask whether
unbounded orbits may exist for outer billiards with less smooth
boundaries. This question has now been answered for a number of
non-differentiable outer billiards.
In the search for outer billiards with unbounded orbits, polygons
are a natural candidates since they are in some sense furthest from
being integrable. In the late 80’s several authors (Shaidenko, Vivaldi,
1987; Kolodziej, 1990; Gutkin, Simanyi, 1992) independently
proved that for a certain class of polygons, extending the class of
lattice polygons with non-parallel sides, the orbits remain bounded.
This larger class, named the quasi-rational polygons, also includes
all regular polygons. At this point the progress on the problem
stalled for a number of years. While numerical studies suggested
that unbounded orbits existed for some polygonal outer billiards as
well as for some other non-smooth shapes rigorous proofs could
not be obtained.
In the last few years there has been a resurgence of interest in
the problem and a positive answer has been discovered in at least
two cases. Specifically, R. Schwartz proved that a large class of
kite-shaped quadrilaterals, including the kite-shaped quadrilateral
of the Penrose tiling do indeed have unbounded orbits (Schwartz,
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2007). Schwartz’s work is deep and far reaching, and is likely to
be only the beginning of a very fruitful line of research. Shortly
thereafter Dmitri Dolgopyat and Bassam Fayad proved existence
of unbounded orbits for a semicircular outer billiard (Dolgopyat,
Fayad, 2007), which were first observed experimentally by
SergeTabachnikov. Thus based on this admittedly small and biased
sample unboundedness appears to prevail for non-smooth billiards.
figure 1.
definition of the outer billiard map.
Surprisingly, for a large subset of trapezoids, which are among the
simplest non-quasi-rational polygons, all outer billiard orbits are
bounded. This is the main result of this announcement.
Outer billiards
Let
be a convex set with oriented boundary. The outer
by taking the image of
billiard map is defined on
Genin
ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
to be the point along the ray tangent to the boundary of , at
the same distance from the point of tangency as p (see Figure 1).
The resulting map has many remarkable properties (see Tabachnikov,
1995, for a good survey).
We will consider outer billiards in which is a trapezoid whose
.
vertices are (-1/2,0), (1/2,0), (1/2,1), (-1/2,1-a), where
Let this set of quadrilaterals be denoted by .
The main result is:
Theorem 1. All orbits of an outer billiard map about a trapezoid in
are bounded.
The result is proved by showing that the first return map to a
special set of lines, forming a Poincaré section of , is symbolically
conjugate to a simple dynamical system related to a circle rotation
. In fact, after the conjugacy is established much more
by
can be said about the orbit structure of the outer billiard map.
For example, periods of all periodic orbits can be easily computed
as well as estimates on the growth of orbit complexity (Genin,
2005).
We now briefly describe the construction leading to the proof of
Theorem 1.
First-return map
We begin by defining a Poincaré section of . The square of the
outer billiard map is a translation by twice the vector joining the
vertices “visited” by the orbit segment.
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figure 2.
two families of invariant line
{x = 2k} and {x =2k + 1}
preserved by T2.
Considering all the possible translation vectors for
it is easy to
see that all of them have a horizontal component either 0 or 2.
is preserved by
Thus the set of lines
(Figure 2). Furthermore, because the orbit winds around the table
in the same direction it can be shown that the first-return map on
every line in this family is well-defined.
By a symmetry argument, the attention can be further restricted
,
. The strip formed
to half-lines
gives a Poincaré section
by the union of the half-lines
of . For simplicity we present the argument for the half-line
, since computations for ,
, are virtually
identical.
Let the first-return map to be denoted by . Since is a piecewise
isometry, is an interval exchange on infinitely many intervals. A
direct but lengthy inspection yields and explicit formula for .
Genin
ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
where
(2)
figure 3.
interval exchange partition
and
The intervals
partition
and are in turn partitioned into
which are exchanged by (Figure 3).
subintervals
the arrangements of subintervals
are clearly
For rational
and their
periodic. Figure 4 shows one cycle of intervals
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corresponding subintervals
laid out horizontally to show
the relationship between partitions of adjacent intervals. The
subinterval highlighted in thick horizontal lines is defined by the first
in (Genin, 2005). The partition
of the two inequalities defining
defined by the second set of inequalities in (Genin, 2005) (indicated
in the figure with double vertical lines) is fixed relative to , while
for each successive interval .
the former interval is shifted by
the partition
repeats with period . For
Thus, for
, on the other hand, the partition
is quasi-periodic.
Figure 4 also shows why for α ∈ all orbits must be bounded:
is capped above and below by
every cycle of the partition
“impassible” intervals, such that orbits from below cannot go up
and orbits from above cannot go down. The reason for this is that
and the interval at the
the interval at the top of the cycle has
,
bottom of the cycle has
transforms to
.
which under
Simple as it is, this is a new observation because parallel sides
exclude the trapezoids from the set of quasi-rational polygons to
which all of the known boundedness results are restricted.
figure 4.
intervals Ik rotated to horizontal
position and stacked vertically, and a
representative orbit (grey lines with
arrows) for a = 1/10
Genin
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ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
Combining this with discreteness of orbits we can conclude that:
Lemma1.
then all orbits are bounded and periodic.
Really, almost any question (e.g. orbit periods, orbit diameters) one
might care to ask about a lattice trapezoid can be answered by
examining a diagram like Figure 4.
A simple consequence of (Dologopyat, Fayad, 2007) is that an
when taken modulo 2.
F-orbit projects to an orbit of rotation by
This observation, which is key to the proof of orbit boundedness,
immediately gives:
Lemma2.
then every non-fixed point of F is non-periodic
That is for irrational only the simplest periodic orbits, that close
after one trip around the table, exist.
Symbolic coding and the conjugate circle rotation
model
The orbit of a point under iterates of F can be coded as a
symbol to be
sequence of 0’s, 1’s and -1’s by taking the
. This encodes how the F-orbit of
moves between the intervals . In spite of the three symbols
the sequence actually turns out to be determined
appearing in
by a binary coding of a circle rotation by .
be a rotation and
Definition 1.
subinterval then the rotation sequence corresponding to
, where is the characteristic function of .
a
is
Following the observation made at the end of Section 3 we can
and the F-orbit coding sequence
relate the rotation sequence
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Lemma3.,
,
and let R be a rotation by
on a circle of length 2 with
. Then the sequence obtained by deleting 1’s
is the same as the sequence
, and the sequence
from
is the same as
.
obtained by deleting -1’s from
Note that whether end points of J are included or excluded is
irrelevant because they correspond to discontinuity points of F.
define a pair of projections from to S1
The two maps and
(of length 2). As Figure 5 illustrates, F-orbits project to rotation
via and
in two different ways. (Notice that the
orbits by
figure also shows how the orbit reenters the lower “impassible”
interval of the cycle, as described above). The “union” of the
corresponding rotation sequences is the coding sequence of the
F-orbit. How these sequences combine to make is determined
by the following scheme. Let S be the circle of length 2,
In words, we move the two points and
around the circle by
iterates of R until one or the other enters the distinguished interval
J (the thick horizontal lines interval in Figure 4) at which point the
point that is outside of J stops while the point inside J continues
to move by R. Once the point that was inside J exits the interval,
the joint motion resumes. If both points enter the distinguished
segment simultaneously has “the right of way”. Of course, the
respectively.
two points and are images of F(x) under and
can be reconstructed by
Consequently, the coding sequence
writing 1’s and -1’s, according to the order in which the points
and visit the interval J.
Orbit boundedness
It is now easy to show that all orbits are bounded. It suffices to
check that
(4)
.
Genin
ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
The proof of this bound proceeds in two steps.
First, we show that the two rotation sequences of Lemma 3
differ only by a shift. This follows easily because
comprising
in (Gutkin, Simanyi, 1992) lie along the same
the points and
rotation orbit.
Next, we need to show that the interleaving of the two rotation
cannot create arbitrarily large
sequences comprising
deviations in the sum (Kolodziej, 1990). Suppose first that the
,
elements of the two rotation sequences simply alternated in
then clearly the supremum in (Kolodziej, 1990) would be at most 1
because 1’s and -1’s cancel almost exactly.
figure 5.
projections p1 (a) and p−1 (b)
This, of course, is not very likely to happen but if we can show
that the corresponding members of the two sequences are never
too far apart the result would clearly follow. A simple analysis of
scheme (Gutkin, Simanyi, 1992) shows this to be the case and we
have:
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,
where C(x) is a constant depending only on x.
To complete the proof we show that the right side of (Moser,
1973) is in turn bounded above by:
The left hand side is a constant depending only on x and Theorem 1
follows. Note that the bound is not uniform in x and so it is possible
for orbits to have arbitrarily large excursions.
Conclusion
A natural extension of this work would be to tackle the case of more
general polygons with families of invariant lines, whose dynamics
can be reduced to an infinite interval exchange on a finite union
of lines. Schwartz succeeded in doing exactly this for kite-shaped
quadrilaterals (Schwartz, 2007). Already in this case, however, the
methods required are considerably more complicated because the
first return map is more complex. The reward for the hard work
are deep and beautiful results (and pictures). One can only guess
what fascinating new challenges and unexpected gems Moser’s
problem will produce.
References
Dologopyat Dmitri, Fayad Bassam (2007), Unbounded orbits for semicircular outer
billiard , (preprint)
Genin Daniel (2005), “Regular and Chaotic Dynamics Of Outer Billiards” Penn State,
[Ph.D. thesis]
Gutkin Eugene, Simanyi Nandor (1992), Dual polygonal billards and necklace
dynamics , “Communications in Mathematical Physics”, V. 143, n. 2, pp. 431-449
Kolodziej Reinhold (1990), The antibilliard outside a polygon, “Bulletin of the Polish
Academic of Sciences Mathematics”, V. 37, pp. 163-168
All URLs checked
December 2009
Genin
ReseaRch announcement: boundedness of oRbits foR tRapezoidal outeR billiaRds
Moser Jürgen (1973), Stable and Random Motions in Dynamical Systems, with
special emphasis on celestial mechanics , Hermann Weyl Lectures, the Institute for
Advanced Study, Princeton, N.J. Annals of Mathematics Studies, n. 77, Princeton
University Press, Princeton, N. J., University of Tokyo Press, Tokyo
Neumann Bernhard (1959), Sharing ham and eggs, Iota, Manchester University
Schwartz Richard (2007), Unbounded orbits for outer billiards , “Journal of Modern
Dynamics”, V. 1, n. 3, pp. 371-424
Shaidenko Anna, Vivaldi Franco (1987), Global stability of a class of discontinuous
dual billiards , “Communications in Mathematical Physics”, V. 110, n. 4, pp. 625-640
Tabachnikov Serge (1995), Billiards , Panoramas et Synthèses 1, Mountrouge, Societé
Matemathique de France
Sintesi
Nella seconda metà degli anni ‘50 Bernhard Neumann introdusse il concetto dell’outer
billiard. Come il nome stesso suggerisce, l’outer billiard è costituito dall’area esterna
ad un biliardo tradizionale, che si ottiene eliminando dall’intero piano cartesiano
una sua porzione limitata di forma qualsiasi. Una volta definite le regole di rimbalzo
all’interno di un biliardo siffatto, si possono studiare le caratteristiche delle orbite
seguite da una bilia analogamente a quanto succede per un biliardo tradizionale.
Ci si può domandare, ad esempio, se ne possono esistere di periodiche e, nel caso,
sotto quali condizioni queste si manifestino.
La differenza fondamentale tra i due tipi di biliardi è data dalla zona all’interno della
quale la bilia può rotolare, essendo questa limitata per il tipo classico e illimitata nel
caso degli outer. Questa diversità è all’origine della domanda che si pose Neumann a
metà degli anni ‘50, e che non ebbe seguiti di rilievo fino alla prima metà degli anni
‘70 quando fu ripresa da Jurgen Moser (motivo per cui tale interrogativo prende il
nome di Domanda di Moser-Neumann): possono esserci orbite illimitate in un outer
billiard? E se sì, sotto quali condizioni?
Tali condizioni sono dettate principalmente da un fattore: la forma del biliardo.
Quest’ultima infatti si è evoluta partendo da quella classica rettangolare fino ad
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assumere le forme più svariate (poligoni, aquiloni di Penrose, semicerchi e aree
delimitate da linee curve chiuse).
Dalla metà degli anni ‘70 ad oggi, i contributi su questo problema sono stati, sotto
un certo aspetto, imprevisti. Inizialmente infatti tali risultati portavano a concludere
che le bilie, per tutte le forme studiate, restassero in area circoscritta nonostante
l’infinito spazio a disposizione. La congettura quindi che si è andata formando
indicava una risposta negativa alla domanda di Moser-Neumann: non esistono orbite
illimitate. Negli ultimi anni, si è dimostrato tuttavia come in presenza di outer-billiard
di particolari forme (quali aquiloni di Penrose o semicerchi) si ottengano orbite
illimitate. In altre parole, l’idea che si era delineata è risultata errata. In fondo,
l’affidabilità di una congettura è basata sul numero di risultati che la confermano.
Ma resta comunque un’ipotesi e, in quanto tale, può essere dimostrata teoricamente
o contraddetta da un controesempio.
Il risultato ottenuto da Daniel Genin si riferisce al momento di rottura tra la congettura
e i risultati che l’hanno contraddetta, essendo, ad ora, l’ultimo che la verifica. La
sua importanza risiede sia nel tentativo riuscito di ampliare la classe di forme che
generano orbite limitate, sia nel fatto che permette una semplice implementazione
informatica del quesito in esame. In particolare, quest’ultima risulta doppiamente utile
poiché da una parte consente di generare e disegnare una grande quantità di orbite
le quali, tramite la loro visualizzazione, sono d’aiuto nel farsi un’idea del risultato
e del come conseguirlo (allontanandosi così dall’idea alla base del bourbakismo);
dall’altra fornisce una descrizione intuitiva del problema sotto studio, rendendolo così
uno strumento efficace dal punto di vista didattico.
Inoltre si potrebbero definire interessanti analogie tra l’approccio qui utilizzato e
altre branche della matematica, quali la meccanica celeste e l’analisi complessa.
Nella analisi proposta dall’autore, si possono infatti cogliere delle somiglianze con le
traiettorie (i.e. orbite) seguite dai corpi celesti immersi in un campo gravitazionale
generato da più masse e con il relativo effetto fionda; così come col percorso (i.e.
orbite) tracciato da un punto nel piano complesso all’iterare di una funzione olomorfa,
come approfondito da Gaston Maurice Julia nel suo lavoro pionieristico sui frattali.
Si è di fronte ad una matematica relativamente giovane (legata, tra l’altro, al teorema
KAM) in cui i diversi risultati ottenuti non hanno ancora portato ad una teoria unitaria.
Tuttavia, volgendo lo sguardo all’interno della storia della matematica, non è difficile
immaginare che in tempi più o meno lunghi ci si troverà con la definizione di una
classe di forme SJB (e di un relativo criterio caratterizzante CJB) l’appartenenza alla
quale (e la soddisfazione del quale) garantirà o meno la presenza di orbite limitate.
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