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**Extra resources for Proceedings of International congress of mathematicians**

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There, it was implicitly assumed that the braid has one free strand. The present statement is correct for arbitrary numbers of strands. 24 Robert Ghrist the appropriate amount. A homotopy theoretic version could be defined in terms of spectra via suspensions. This, then, yields a formal index for arbitrary (proper) braid pairs. The real question is what dynamical meaning this generalized index entails. The passage from positive braids to arbitrary braids is akin to the passage from a Lagrangian to a Hamiltonian settings, and such an extended index appears to be a relative Floer homology for (multiply) periodic solutions to time-periodic Hamiltonian systems.

Then the sum over n of (2) gives an upper bound μ n Bn (c ) ≤ n≥1 μn (c ) on the set of all parameters ε for which fε has a periodic point of some period n that is not (n, cγn )-hyperbolic. If the sum converges and n≥1 μn (c ) = μ(c) → 0 as c → 0, then for μ-almost every ε there is c > 0 such that for every n every periodic point of period n is (n, cγn )-hyperbolic. This statement (almost) implies that all periodic points of period n are at least ≈ cγn -apart and, therefore, the number of periodic points is bounded by ≈ (cγn )− dim M 2 In [KH1] we use a stronger property of hyperbolicity of periodic points (see Section 2 of that paper).

For the 1-dimensional Lebesgue measure of the u0 ’s we have Leb {u0 : |fu (x0 ) − x1 | = |f (x0 ) + u0 − x1 | ≤ γ˜n } ≤ 2γ˜n . Fix u0 . Similarly, the position of fu (x1 ) depends only on u1 (see (4) and Figure 1). Thus, we have Leb u1 : |fu (x1 ) − x2 | = |f (x1 ) + u0 + u1 (x1 − x0 ) − x2 | ≤ γ˜n } ≤ 2γ˜n . |x1 − x0 | Inductively for k = 2, . . , n − 1, fix u0 , . . , uk−1 . Then the position of fu (xk ) depends only on uk . Moreover, for k = 2, . . , n − 2 we have k Leb uk : |fu (xk ) − xk+1 | = f (xk ) + m−1 um m=0 ≤ 2γ˜n k−1 j =0 |xk − xj | (xk − xj ) − x0 ≤ γ˜n j =0 , 35 Newton interpolation polynomials and discretization method and for k = n − 1 we have Leb un−1 : |fu (xn−1 ) − x0 | ≤ γ˜n ≤ n−2 j =0 2γ˜n .