the connecting lemma(s) following hayashi, wen&xia, arnaud
TRANSCRIPT
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The Connecting Lemma(s)
Following Hayashi, Wen&Xia, Arnaud
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Pugh’s Closing Lemma
• If an orbit comes back very close to itself
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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 ?
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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 ?
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An orbit coming back very close
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A C0-small perturbation
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The orbit is closed!
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A C1-small perturbation: No closed orbit!
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For C1-perturbation less than , one need a safety distance, proportional to the jump:
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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)
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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
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For diffeos
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2) Selecting points:
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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?
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The connecting lemma (Hayashi 1997)
By a C1-perturbation:
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Variations on Hayashi’s lemma
x non-periodic point
Arnaud,Wen & Xia
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Corollary 1: for C1-generic f,H(p) = cl(Ws(p)) cl(Wu(p))
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Other variation
x non-periodic
in the closure of
Wu(p)
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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
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30 years from Pugh to Hayashi : why ?
Pugh’s
strategy :
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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.
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Hayashi changes the strategy:
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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.
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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
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Then the cube C verifies:
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For any pair x,y
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There are x=x0, …,xN=y such that
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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 )
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Perturbation boxes1) Tiled cube : the ratio between adjacent tiles is bounded
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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
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There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
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There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
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There is g -C1-close to f,
perturbation in Cf(C)…fN-1(C)
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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
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Why this statment implies the connecting lemmas ?
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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
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For definition of perturbation box, there is a g C1-close to f
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Proof of the connecting lemma:
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Consider (xi,yi) in the same tile
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Consider the last yi in the tile of x0
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And consider the next xi
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Delete all the intermediary points
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Consider the last yi in the tile
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Delete all intermediary points
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On get a new sequence (xi,yi) with at most 1 pair in a tile
x0 and yn
are the original
x0 and yn
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Pugh gives sequences of points joining xi to yi
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There may have conflict between the perturbations in adjacent tiles
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Consider the first conflict zone
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One jump directly to the last adjacent point
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One delete all intermediary points
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One does the same in the next conflict zone, etc, until yn
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Why can one solve any conflict?
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There is no m other point nearby!the strategy is well defined
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