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The Connecting Lemma(s) Following Hayashi, Wen&Xia, 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 C 0 -small perturbation

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The orbit is closed!

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A C 1 -small perturbation: No closed orbit!

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For C 1 -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 C 1 -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 C 1 - small perturbations?

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The connecting lemma (Hayashi 1997) By a C 1 -perturbation:

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Variations on Hayashis lemma x non-periodic point Arnaud, Wen & Xia

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Corollary 1: for C 1 -generic f, H(p) = cl(W s (p)) cl(W u (p))

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Other variation x non-periodic in the closure of W u (p)

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Corollary 2: for C 1 -generic f cl(W u (p)) is Lyapunov stable Carballo Morales & Pacifico Corollary 3: for C 1 -generic f H(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 argumentFor any C 1 -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 C f i (C)= Ø 0*
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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=x 0, …,x N =y such that

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The ball B( f i (x i ), d(f i (x i ),f i (x i+1 )) ) 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 (x 0,y 0 ), …, (x n,y n ), with x i & y i in the same tile

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There is g -C 1 -close to f, perturbation in C f(C) … f N-1 (C)

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There is g -C 1 -close to f, perturbation in C f(C) … f N-1 (C)

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There is g -C 1 -close to f, perturbation in C f(C) … f N-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, 0*
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*

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Why this statment implies the connecting lemmas ?

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x 0 =y 0 =f i(0) (p) x 1 =y 1 =f i(1) (p) … x n =f i(n) (p); y n =f –j(m) (p) x n+1 =y n+1 =f -j(m-1) (p) … x m+n =y m+n =f –j(0) (p) By construction, for any k, x k and y k belong to the same tile

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For definition of perturbation box, there is a g C 1 -close to f

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Proof of the connecting lemma:

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Consider (x i,y i ) in the same tile

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Consider the last y i in the tile of x 0

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And consider the next x i

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Delete all the intermediary points

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Consider the last y i in the tile

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Delete all intermediary points

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On get a new sequence (x i,y i ) with at most 1 pair in a tile x 0 and y n are the original x 0 and y n

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Pugh gives sequences of points joining x i to y i

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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 y n

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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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