The periodic orbits of an area preserving twist map
Summary (1 min read)
C F(C)
- There are several structures on X which are relevant for their problem.
- Hence all the off diagonal elements of the Jacobian of grad W are nonnegative, which implies (a).
- For parabolic equations this has been generalized by Matano [9] .
- The following result was proved by Smillie [13] : Proposition 2.3.
Proposition 3.2. The spectrum of L is given by
- This set (with the appropriate topology) is naturally identified as a circle ("polar coordinates"), and therefore the authors may regard M(λ) as an orientation preserving diffeomorphism of the circle.
- The orientation explains the occurrence of the +1 in the Jordan normal forms for some μ >.
- In this way one verifies the classification of the M(A)'s given above.
Definition. τ(s) = -ρ(0).
- As maps of the circle the integer part of their rotation numbers are not defined, and it is this integer part which the authors shall need.
- A straightforward calculation shows that this flow is gradientlike with respect to.
V(ξ) = (ξ,Lξ).
- For the particular Jacobi matrix one gets by putting ^ΞI and β { = 0, one can explicitly determine the I k and their homotopy indices.
- The reader who wants to verify this should bear in mind that, since the flow is gradient like, all isolated invariant sets consist of fixed points and orbits connecting them.
- The authors claim that the I k with \<2k<q contain at least two such pairs.
- After some thought one concludes from this that: and if q is even, which proves the lemma.
The final step in computing h
- Apply the construction (and notation) of Sect, three to this situation.
- Using the fact that the periods of the solutions of X"(t) + ω 2 /πsmπX(ή = O increase with amplitude one sees that the matrix M(0) is conjugate to Ίwith α>0.
Corollary. h(M k ) = h k is the homotopy type of S 1
- By the unstable manifold theorem the flow near M k is the product of that near a hyperbolic fixed point with index 2/c -1, and the trivial flow on a circle.
- This corollary finishes the proof of Theorem 5.
Lemma 10.3. There is an orbit y in Y such that a<y<b and y -x has one or two sign changes
- The authors can exclude the first alternative for the following reasons.
- Thus the authors are left with the second alternative, which proves the lemma.
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"The periodic orbits of an area pres..." refers background in this paper
...For such a diffeomorphism the rotation number is defined up to an integer (see Coddington and Levinson [3]), and it depends continuously on the mapping, i....
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1,662 citations
"The periodic orbits of an area pres..." refers methods in this paper
...We shall assume the reader to be familiar with the Morse-Conley index as it is described in Conley [4], or in Conley and Zehnder [5, 6]....
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...Now use the product rule to obtain h(Mk) (see Conley [4])....
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607 citations
"The periodic orbits of an area pres..." refers background or methods in this paper
...The generating function h is uniquely determined (up to a constant) by the map F. Its construction is given by Mather [10]. See also Aubry and le Dacron [ 2 ]....
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...We refer to Mather [10, 111 and Aubry and le Dacron [ 2 ] for more details....
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511 citations
"The periodic orbits of an area pres..." refers background in this paper
...The generating function h is uniquely determined (up to a constant) by the map F. Its construction is given by Mather [ 10 ]....
[...]