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with the Birkhoff periodic orbits of type p/q, if
the phase-locking interval 
is nontrivial.
Birkhoff periodic orbits and Aubry-Mather sets.
Denote by 
a lift of 
to
and let R be the map
.
A periodic orbit is called a (p,q)-Birkhoff periodic orbit, if
there exists a continuous function 
such that 
.
It follows from a basic
fact for monotone twist maps that for any 
at least two Birkhoff periodic orbits of type (p,q) exist
[18, 15, 11].
One of these orbits is a maximum of the
functional |O|, the total length of the trajectory O.
Let 
be the set of Birkhoff periodic orbits.
Given 
irrational. An accumulation point
(in the Hausdorff topology) of sets
as 
is called an
Aubry-Mather set and denoted by 
. Such a set
has the property that it is the graph of a Lipschitz continuous function
defined on some compact subset K
of 
and such that a lift of
preserves the order of the covering of M [15].
Typically, Birkhoff periodic points are isolated. The case
of billiards shows however that one has to deal in general with
whole arcs of Birkhoff periodic points.
If a connected periodic set Y
contains a Birkhoff periodic point, then every point in Y is a Birkhoff
periodic point and we will call a connected component
a Birkhoff periodic set. In a typical situation, a
Birkhoff periodic set is a Birkhoff periodic point.
The set of Birkhoff periodic points is isolated from the rest of
(p,q)-periodic orbits.
The index of a periodic set.
Let C be a simple, contractible oriented closed curve in 
avoiding all
fixed points of 
. The index 
of C with respect to the fixed points of is defined as
the Brower degree of the map 
,
,
where 
is calculated in a chart.
The index does not change if we deform C without intersecting
a fixed point of .
The index is also a homotopy invariant: if 
is a family of maps
and for all 
in some parameter interval I and C is disjoint from the
fixed points of , then 
is constant for 
.
If a curve C contains only one fixed point 
of , then
is called the index of . If C contains
a connected fixed-point set 
, we call
the
index of this fixed point set.
If a curve C contains finitely
many fixed points 
(rsp. connected fixed-point sets 
)
of , then 
(see for example [5] Theorem 14.4.4 or [25]).
Examples. Assume C contains only
one hyperbolic periodic point of period q,
then 
.
If C contains one elliptic periodic point, then
.
If C contains no periodic point, then 
.
More generally: if is differentiable and for a fixed point
of the matrix 
has no eigenvalue 1, then
has index 1 if 
and index
-1 if 
.
(See [23] Lemma 4 or [28] Proposition 3.). The index can
geometrically also be determined by
Poincaré's index formula (see [28]).
The index of a fixed point of an area-preserving
homeomorphism is bounded above
by 1 [28] [25].
From this follows by a generalized Poincaré-Hopf's theorem [27]
that if the induced homeomorphism 
on 
has n attracting periodic intervals and so n repelling
intervals on
(they all must have index -1), then there exist
n sets of index 1 of the same period on each side of .
Remark. A periodic orbit of a differentiable
monotone twist map is a critical point
of a functional. If the critical point is nondegenerate, then the orbit
is either elliptic or hyperbolic. The Morse-index, the number of negative
eigenvalues at a critical point and the index 
are related by 
. The reason is that a
nondegenerate hyperbolic orbit has even Morse-index and a nondegenerate
elliptic orbit has odd Morse-index (see [32]).
Bifurcations of sets of periodic orbits.
We consider a one-parameter
family of monotone twist maps 
parameterized by points in some interval I.
Let C be a homotopically trivial
simple closed curve in such that for ,
no fixed point of 
is on C and such that C is
the boundary of an open subset 
of .
Since 
is bounded away from 0
and 
Equation 2), there are only
finitely many connected fixed point sets in .
A parameter value , for which
the number of connected components of fixed points of of
in
changes, is called a bifurcation point. (Usually,
a bifurcation point is a parameter point,
for which the topological type of the fixed point set
inside C changes. Since here, only the number
of components will be relevant, we take the narrower definition.)
Index considerations limit the possibilities for bifurcations
of periodic orbits in monotone twist maps.
Examples.
1) Assume, for each , there exists at least one fixed
point 
of in .
Assume 
is hyperbolic for 
, parabolic for 
and elliptic for 
. If is the only fixed point in
for , then 
is a bifurcation value
because the index of
changes from -1 to 1 and the sum of the indices
of all fixed points of in stays constant.
If two additional fixed points
of index -1 are created at , then the bifurcation is
a pitchfork bifurcation.
2) Assume, a single fixed point exists in
for , and does not
exist for . Then it must have index 0 and must be parabolic.
3) Also the reversed processes are possible: for example,
two fixed points of index -1 collide with a fixed point of index 1
and leave a fixed point of index -1, or
three fixed points, two of them with index -1
and one with index 1, collide for leaving a lonely
fixed point of index -1.
Remarks.
1) If all periodic orbits of rotation number p/q are isolated, then
at least two Birkhoff periodic orbits have index different from 0 because
any periodic orbit of index 0 can be removed by a local perturbation
of the map [30].
2) The bifurcations of periodic points of a generic one-parameter family of
area-preserving maps of a surface has been classified in [22].
Passage of Aubry-Mather sets through the invariant circle.
As the parameter varies, the invariant
circle belonging to the caustic 
moves up in the phase space .
At each parameter , the invariant circle divides
the phase space into a region below bounded
by 
and and
a region above .
For small 
, the invariant circle
is near the boundary component
of the phase space .
For 
, the invariant circle
approaches the equator 
of .
Aubry-Mather sets with a fixed
rotation number pass through the moving circle .
The passage of
with irrational is easy to describe: since each parameter value
with irrational
is a point of increase of 
, the set of parameters
for which
intersects with consists of
exactly one point, the parameter value 
,
for which 
.
The set is in general a Cantor set.
If the map and the curve are smooth and satisfies a
Diophantine condition, then more can be said:
both and
depend in a differentiable way on
(see [4]). If is smooth and is
a smooth invariant curve with Diophantine rotation number , then
is accumulated by other invariant curves filling a
set of positive measure [33].
Therefore, if 
is Diophantine, the moving circle
is accumulated by invariant curves which fill out a set of
positive measure.
We analyze, how M(p/q) which consists of
Birkhoff periodic orbits passes through if
the phase locking interval is nontrivial.
Since any periodic orbit on is a Birkhoff periodic orbit,
is the set of parameters for which M(p/q) intersects
.
The picture is that for a residual set of caustics and
a dense set of parameters 
, every Birkhoff periodic
orbit on the invariant curve is
isolated, hyperbolic and has index -1. Since these orbits correspond
to periodic orbits of the circle map , we can give them
coordinates 
and group them in
pairs 
.
(Compare Figure 11a). Important for us is the following Corollary.
Every q-periodic orbit 
of the billiard map is determined by the values
and 
is a
critical point of the
length functional 
, which is the length
of the polygon.
Note. We used here again the notation 
, 
but for different things.
Since there is no danger of confusion and this notation is common for the
entries of a Jacobi matrix.
The passage of Birkhoff periodic sets of a given rotation number p/q is in general a complicated sequence of possibly simultaneous bifurcations. There are two basic building blocks, the passage of an index -1 set and the passage of an index +1 set. Both events are bifurcations and the second event is the time reversed process of the first one. We first look at the first rsp. last bifurcation value.
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, if there is no invariant circle with
rotation number . For 
, the non-Birkhoff periodic
points can only be above and for
, they can exist only below . On the other
hand, they can not collide with . Thus, the
non-Birkhoff periodic orbits appear and disappear in the course of
the deformation. sufficiently near to 
,
an isolated hyperbolic Birkhoff periodic orbit
of index -1 approaches the caustic, hits the caustic as a parabolic
orbit of index -1 and leaves the caustic as an isolated elliptic
orbit with index 1. At the point , two hyperbolic
orbits of index -1 are created and move
along the caustic. They might interact with other orbits of index -1
and each of those events is accompanied with a passage of a Birkhoff
periodic orbit through the caustic as described in
Theorem 5.6. At the end of the interval
, the last Birkhoff periodic orbit of index 1 passes
through the caustic, changing from elliptic to parabolic
and leaves it hyperbolic with index -1. 
,
then all the Birkhoff periodic sets hit the curve at the
same time 
and is exceptional
consisting of periodic points. have derivatives bounded away from
zero and , there are only
finitely many bifurcation points in 
.
There are the following conservation laws for collisions of
periodic sets with the invariant curve belonging to the caustic:
to each orbit with index 1 hitting the curve
for some , there is
a set of index 1 leaving for some 
.
The same is true for orbits with index -1. Thus,
the sum of the indices of Birkhoff periodic orbits
passing through the caustic curve is equal to zero. Next: On this document... Up: Billiards that share a Previous: Periodic orbits and
Oliver Knill
