A Brouwer-like theorem for orientation reversing homeomorphisms of the sphere

We provide a topological proof that each orientation reversing homeomorphism of the 2-sphere which has a point of period k greater than or equal to 3 also has a point of period 2. Moreover if such a k-periodic point can be chosen arbitrarily close to an isolated fixed point. o then the same is true for the 2-periodic point. We also strengthen this result by proving. that if an orientation reversing homeomorphism h of the sphere has no 2-periodic point then the complement of the fixed point set can be covered by invariant open sets where h is conjugate either to the map (x, y) --> (x + 1, -y) or to the map (x, y) --> 1/2(x. -y).