The Connecting Lemma(s)

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The Connecting Lemma(s)

• Following Hayashi, WenXia, Arnaud

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Pughs Closing Lemma

• If an orbit comes back very close to itself

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Pughs 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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Pughs 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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Pughs 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 (PughRobinson)

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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 Hayashis 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 ?
Pughs 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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Hayashis 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 Pughs argument
For any C1-neighborhood of f and any ?gt0 there
is Ngt0 such that
For any point x there are local coordinate
around x such that
Any cube C with edges parallela to the axes
and C?f i(C) Ø 0lti?N
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Then the cube C verifies
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For any pair x,y
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There are xx0, ,xNy such that
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The ball B( f i(xi), ? d(f i(xi),f i(xi1)) )
where ? is the safety distance
is contained in f i( (1?)C )
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Perturbation boxes

• 1) 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 C?f(C)??fN-1(C)

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There is g ?-C1-close to f, perturbation in
C?f(C)??fN-1(C)
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There is g ?-C1-close to f, perturbation in
C?f(C)??fN-1(C)
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The connecting lemma

• Theorem Any tiled cube C,
• whose tiles are Pughs tiles
• and verifying C?f i(C) Ø, 0lti?N
• is a perturbation box

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Why this statment implies the connecting lemmas ?
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x0y0f i(0)(p) x1y1f i(1)(p) xnf i(n)(p)
ynf j(m)(p) xn1yn1f -j(m-1)(p) xmnymn
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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