The necessity of the star-shaped condition on Ding’s version of the Poincar´ e-Birkhoff theorem. Rog´ erio Martins ∗ Dep. Matem´atica, FCT, Universidade Nova de Lisboa Monte da Caparica, 2829-516 Caparica, Portugal. E-mail: roma@fct.unl.pt Antonio J. Ure˜ na † Dep. Matem´atica Aplicada, Facultad de Ciencias, Universidad de Granada Av. Fuentenueva, s.n., 18071, Granada, Spain. E-mail: ajurena@ugr.es Abstract: W-Y. Ding proved a generalization of the Poincar´ e-Birkhoff fixed point Theorem not requiring that the boundary curves of the anular region to be circles and invariant. However the inner boundary is required to be star-shaped with respect to the origin. We construct a counterexample that shows that the star-shaped condition is necessary. 1 Introduction The well known Poincar´ e-Birkhoff theorem states that any area-preserving homeomorphism T of the annulus {(x, y ) ∈ R 2 :0 <R 2 1 ≤ x 2 + y 2 ≤ R 2 2 } to itself, keeping invariant both boundary circles, and rotating them in opposite directions, has at least two fixed points. Poincar´ e stated this theorem [11] and gave a proof for some particular cases. The theorem was subsequently proved in its full generality by Birkhoff [1, 2] (see also [3]). In order to make the theorem more suitable for applications, Ding [7] generalized it, allowing the boundary curves not to be circles or invariant. More precisely, he considered an annular region A bounded by an inner boundary C 1 and ∗ Supported by FCT, POCI/Mat/57258/2004. † Supported by D.G.I. MTM2005-03483, Ministerio de Educaci´ on y Ciencia, Spain. Mathematics Subject Classification: Primary 54H25. 1