The Connecting Lemma(s)
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Transcript of The Connecting Lemma(s)
The Connecting Lemma(s)
Following Hayashi, Wen&Xia, Arnaud
Pugh’s Closing Lemma
• If an orbit comes back very close to itself
Pugh’s Closing Lemma
• If an orbit comes back very close to itself
•Is it possible to close it by a small pertubation of the system ?
Pugh’s Closing Lemma
• If an orbit comes back very close to itself
•Is it possible to close it by a small pertubation of the system ?
An orbit coming back very close
A C0-small perturbation
The orbit is closed!
A C1-small perturbation: No closed orbit!
For C1-perturbation less than , one need a safety distance, proportional to the jump:
Pugh’s closing lemma (1967)
If x is a non-wandering point of a diffeomorphism f on a compact manifold, then there is g, arbitrarily C1-close to f, such that x is a periodic point of g.
•Also holds for vectorfields
•Conservative, symplectic systems (Pugh&Robinson)
What is the strategy of Pugh?
• 1) spread the perturbation on a long time interval, for making the constant very close to 1.
For flows: very long flow boxes
For diffeos
2) Selecting points:
The connecting lemma• If the unstable
manifold of a fixed point comes back very close to the stable manifold
•Can one create homoclinic intersection by C1-small perturbations?
The connecting lemma (Hayashi 1997)
By a C1-perturbation:
Variations on Hayashi’s lemma
x non-periodic point
Arnaud,Wen & Xia
Corollary 1: for C1-generic f,H(p) = cl(Ws(p)) cl(Wu(p))
Other variation
x non-periodic
in the closure of
Wu(p)
Corollary 2: for C1-generic fcl(Wu(p)) is Lyapunov stable
Carballo Morales & Pacifico
Corollary 3: for C1-generic fH(p) is a chain recurrent class
30 years from Pugh to Hayashi : why ?
Pugh’s
strategy :
This strategy cannot work for connecting lemma:
• There is no more selecting lemmas
Each time you select one red and one blue point,There are other points nearby.
Hayashi changes the strategy:
Hayashi’s strategy.
• Each time the orbit comes back very close to itself, a small perturbations allows us to shorter the orbit:
one jumps directly to the last return nearby, forgiving the intermediar orbit segment.
What is the notion of « being nearby »?Back to Pugh’s argument For any C1-neighborhood of f and any
>0 there is N>0 such that:
For any point x there are local
coordinate around x such that
Any cube C with edges parallela to the axes
and Cf i(C)= Ø
0<iN
Then the cube C verifies:
For any pair x,y
There are x=x0, …,xN=y such that
The ball B( f i(xi), d(f i(xi),f i(xi+1)) ) where is the safety distance
is contained in f i( (1+)C )
Perturbation boxes1) Tiled cube : the ratio between adjacent tiles is bounded
The tiled cube C is a N-perturbation box for (f,) if:
for any sequence (x0,y0), … , (xn,yn),
with xi & yi in the same tile
There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
The connecting lemma
Theorem Any tiled cube C,
whose tiles are Pugh’s tiles
and verifying Cf i(C)= Ø, 0<iN
is a perturbation box
Why this statment implies the connecting lemmas ?
x0=y0=f i(0)(p)x1=y1=f i(1)(p)…xn=f i(n)(p); yn=f –j(m)(p)xn+1=yn+1=f -j(m-1)(p)…xm+n=ym+n=f –j(0)(p)
By construction, for any k,
xk and yk belong to
the same tile
For definition of perturbation box, there is a g C1-close to f
Proof of the connecting lemma:
Consider (xi,yi) in the same tile
Consider the last yi in the tile of x0
And consider the next xi
Delete all the intermediary points
Consider the last yi in the tile
Delete all intermediary points
On get a new sequence (xi,yi) with at most 1 pair in a tile
x0 and yn
are the original
x0 and yn
Pugh gives sequences of points joining xi to yi
There may have conflict between the perturbations in adjacent tiles
Consider the first conflict zone
One jump directly to the last adjacent point
One delete all intermediary points
One does the same in the next conflict zone, etc, until yn
Why can one solve any conflict?
There is no m other point nearby!the strategy is well defined