By Gravesa W. G., Peckhamb B., Pastorc J.

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Preprint, 2005. , Morse theory on spaces of braids and Lagrangian dynamics. Invent. Math. 152 (2003), 369–432. , Topological mixing with ghost rods. Preprint, 2005. [34] M. Hirsch, Systems of differential equations which are competitive or cooperative, I: Limit sets. SIAM J. Math. Anal. 13 (1982), 167–179. , Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences. Arch. Rational Mech. Anal. 90 (2) (1985), 115–193. , Closed characteristics of second order Lagrangians.

Our plan is to verify that for most parameters this cone condition holds after each loop: Dfεni +1 (Kαn (pi )) ⊂ Kαn (pi+1 (mod s) ) for each i = 0, . . , s − 1. (20) This condition clearly implies (19), because the image of the first cone Kαn (p0 ) belongs to the second cone Kαn (p1 ). The image of the second one belongs to the third one and so on. Fix 0 < α 1. Notice that if all loops are long: ni > 3αn, then Lni Kαn (pi ) is the cone of width angle < 2μ−αn . Fix 1 ≤ j ≤ s. To satisfy condition (20) for j 49 Newton interpolation polynomials and discretization method we need to avoid the intersection of the cone Dfε,p˜ j (Lnj Kαn (pj )) and a complement to Kαn (p˜ j +1 ) (see Figure 7 for p = pj +1 ).

In order to fail the inductive hypothesis of order n with constants c , a diffeomorphism fε should have a periodic, but not (n, cγn )-hyperbolic point x = fεn (x). There is a continuum of possible n-tuples {xk }0≤k≤n such that for some ε ∈ B we have f (xk ) = xk+1 (mod n) and x0 is not (n, cγn )-hyperbolic. Instead of looking at the continuum of n-tuples, we discretize this space and consider only those n-tuples {xk }0≤k≤n that lie on a particular grid, denoted Iγ˜n , and replace trajectories by γ˜n pseudotrajectories.