Understanding Pugh’s Closing Lemma and Hayashi’s Connecting Lemma in Dynamical Systems

1. The Connecting Lemma(s) Following Hayashi, Wen&Xia, Arnaud

2. Pugh’s Closing Lemma • If an orbit comes back very close to itself

3. 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 ?

4. 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 ?

5. An orbit coming back very close

6. A C0-small perturbation

7. The orbit is closed!

8. No closed orbit! A C1-small perturbation:

9. For C1-perturbation less than , one need a safety distance, proportional to the jump:

10. Pugh’s closing lemma (1967) Ifx 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)

11. 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

12. For diffeos

13. 2) Selecting points:

14. 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?

15. The connecting lemma (Hayashi 1997) By a C1-perturbation:

16. Variations on Hayashi’s lemma x non-periodic point Arnaud, Wen & Xia

17. Corollary 1:for C1-generic f,H(p) = cl(Ws(p))  cl(Wu(p))

18. Other variation x non-periodic in the closure of Wu(p)

19. 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

20. 30 years from Pugh to Hayashi : why ? Pugh’s strategy :

21. 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.

22. Hayashi changes the strategy:

23. 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.

24. 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 Cf i(C)= Ø 0<iN

25. Then the cube C verifies:

26. For any pair x,y

27. There are x=x0, …,xN=y such that

28. 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 )

29. Perturbation boxes 1) Tiled cube : the ratio between adjacent tiles is bounded

30. 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

31. There is g-C1-close to f, perturbation in Cf(C)…fN-1(C)

32. There is g-C1-close to f, perturbation in Cf(C)…fN-1(C)

33. There is g-C1-close to f, perturbation in Cf(C)…fN-1(C)

34. The connecting lemma Theorem Any tiled cube C, whose tiles are Pugh’s tiles and verifying Cf i(C)= Ø, 0<iN is a perturbation box

35. Why this statment implies the connecting lemmas ?

42. 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

44. For definition of perturbation box, there is a g C1-close to f

45. Proof of the connecting lemma:

46. Consider (xi,yi) in the same tile

47. Consider the last yiin the tile of x0

48. And consider the next xi

49. Delete all the intermediary points

50. Consider the last yiin the tile