Sur les dynamiques holomorphes non linéarisables et une conjecture de V. I. Arnold

— We solve a conjecture of V. I. Arnold by eonstructing a non linearizable analytic diffeomorphism of thé circle for which any analytic extension bas no periodic orbits. We study thé same problem for holomorphic germs/(z)==Xz+^(z), À-^'", aeR-Q.ifthe diophantine condition £ ^"^ogiog^.^+oo BS1 hoids, where {pJq^n^Q are tne convergents of a, we construct a non-linearizable holomorphic genn /(z)==À.z+fi?(z), À.=^", with no periodic orbits except 0. This condition is optimal: Ifit is not satisfied and /is non-linearizable, we prove thé existence of a séquence of periodic orbits of/converging to 0. We study aiso thé case of polynomial germs/(z)=À-z+(P(z). In this context we prove, using a resuit of Yoœoz for thé quadratic polynomial, that when a is not a Brjuno number any polynomial germ P(z)=^z+a2Z+ . . . +0^ structurally stable has a séquence of periodic orbits converging to 0 and so, is non-linearizable.