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But this contradicts the fact that the number of zeros of bn -orbits increases with n. We conclude therefore that limn→∞ bn = 0. This completes the proof of Proposition 1. Returning to the original variables f (ρ) and ρ, and using the notation hn (x) ≡ hbn (x), we have fn (ρ) = hn (x) + π/2 with fn (1) = π/2 and fn (0) = 0(mod π ), as claimed in Theorem 1. We end this section with a remark about the large n limit. From limn→∞ bn = 0, it follows that limn→∞ hn (x) = 0 for any finite x. The limiting solution h∗ = 0 (or f ∗ = π/2) is a singular map which geometrically corresponds to the map into the equator of S 3 .

Before doing it however, we shall prove a bound on the force generated by the configuration Xn (τ ) over the particle i, which will be often used in the sequel. 19) µ 3/2 C9 R(τ )3/2 ≤ C14 R(τ )3/2 . 1. 7), V n (0) ≤ Q(X)1/2 ϕ(n) = Q(X)1/2 R(0) (this determines the dependence of V n (0), and hence V n (t), on n). 1) is verified for t = 0. 36 E. Caglioti, C. Marchioro, M. 20) for a suitable constant A to be fixed later and satisfying A > 2(Q(X)1/2 + 1). 16)). 24) and hence: R 4/6 |t ∗ −t1 | α therefore, for a suitable choice of α ∈ [1/2, 1], Furthermore 1 1 2 ∗ vi (t ) − vi2 (t1 ) = 2 2 = t∗ t1 vi · Fi,j j t∗ t1 ds H ds (vi − vj ) · Fi,j + j ∗ =− t1 +h h=1 t1 +(h−1) ∗ φ(xi (t ) − xj (t )) + j ds vj · Fi,j j φ(xi (t1 ) − xj (t1 )) j H + is integer.

On non-equilibrium dynamics of multidimensional infinite particle systems in the translation invariant case Commun. Math. Phys. : Construction of the dynamics for one-dimensional systems of Statistical Mechanics. Sov. Theor. Math. Phys. : The construction of the cluster dynamics of dynamical systems in Statistical Mechanics. Vest. Moskow Univ. Sez. I Math. Mech. 29, 152–176 (1974) Communicated by Ya. G. Sinai Commun. Math. Phys. 215, 45 – 56 (2000) Communications in Mathematical Physics © Springer-Verlag 2000 Equivariant Self-Similar Wave Maps from Minkowski Spacetime into 3-Sphere Piotr Bizon´ Institute of Physics, Jagellonian University, Kraków, Poland Received: 20 October 1999 / Accepted: 12 May 2000 Abstract: We prove existence of a countable family of spherically symmetric selfsimilar wave maps from 3 + 1 Minkowski spacetime into the 3-sphere.