In this paper we prove that the $S^1$-invariance of the Poincaré action functional associated to the Lorentz force equation gives the existence of multiple critical points which are periodic solutions with a fixed period. To do this, we prove an abstract multiplicity result which is based upon the Lusternik-Schnirelman method with the $S^1$-index. The corresponding result in the context of the Fadell-Rabinowitz index is proved in Ekeland and Lasry (Ann. Math., 112 (1980)). The main feature of our abstract result is that it allows us to consider nonsmooth functionals satisfying only a weak compactness condition well adapted to the Poincaré functional.

We introduce a new functional inspired by Perelman's λ-functional adapted to the asymptotically locally Euclidean (ALE) setting and denoted λ ALE . Its expression includes a boundary term which turns out to be the ADM-mass. We prove that λ ALE is defined and analytic on convenient neighborhoods of Ricci-flat ALE metrics and we show that it is monotonic along the Ricci flow. This for example lets us establish that small perturbations of integrable and stable Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass. We then introduce a general scheme of proof for a Lojasiewicz-Simon inequality on non-compact manifolds and prove that it applies to λ ALE around Ricci-flat metrics. We moreover obtain an optimal weighted Lojasiewicz exponent for metrics with integrable Ricci-flat deformations. ALE 3. A functional defined on a C 2,α τ -neighborhood of Ricci-flat ALE metrics 4. Energy estimates on the potential function 5. Fredholm properties of the Lichnerowicz operator in weighted spaces 6. Properties of λ ALE in the integrable case 7. A Lojasiewicz inequality for λ ALE : the general case 8. Deformation of Ricci-flat ALE metrics, scalar curvature and mass Appendix A. Real analytic maps between Banach spaces Appendix B. Divergence-free gauging of asymptotically conical spaces Appendix C. First and second variations of geometric quantities References

We show that if one selects uniformly independently and identically distributed matrices A 1 , . . . , A s P SL 2 pZq from a ball of large radius X then with probability at least 1 ´X´1`op1q the matrices A 1 , . . . , A s are free generators for a free subgroup of SL 2 pZq. Furthermore, to show the flexibility of our method we do similar counting for matrices from the congruence subgroup Γ 0 pQq uniformly with respect to the positive integer Q ď X. This improves and generalises a result of E. Fuchs and I. Rivin (2017) which claims that the probability is 1 `op1q. We also disprove one of the statements in their work that has been used to deduce their claim. 2020 Mathematics Subject Classification. 11C20, 15B36, 15B52. Key words and phrases. Free group, SL 2 pZq matrices. for an integer Q ě 1. Furthermore, let Γ 0 pQ, Xq " Γ 0 pQq X SL 2 pZ; Xq.