
Birkhoff periodic orbits and AubryMather 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
AubryMather 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 fixedpoint set , we call
the
index of this fixed point set.
If a curve C contains finitely
many fixed points (rsp. connected fixedpoint 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 areapreserving
homeomorphism is bounded above
by 1 [28] [25].
From this follows by a generalized PoincaréHopf'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 Morseindex, 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 Morseindex and a nondegenerate
elliptic orbit has odd Morseindex (see [32]).
Bifurcations of sets of periodic orbits.
We consider a oneparameter
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 oneparameter family of
areapreserving maps of a surface has been classified in [22].
Passage of AubryMather 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 .
AubryMather 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 qperiodic 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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Oliver Knill