Fixed points of periodic maps

Let f be a periodic differentiable map from a sphere to itself. A well-known conjecture of Smith asserts that in many cases (e.g., when the fixed points are isolated) the derivatives of f at its fixed points, regarded as Jacobian matrices, are linearly similar. Here we give counterexamples to this conjecture. The results show that, in many cases, these Jacobian matrices are only nonlinearly similar. This uses our recent discovery of orthogonal matrices which are nonlinearly similar without being linearly similar. Some results on general smooth actions of finite groups on differentiable manifolds are presented; the topological equivalence of their tangential representations at the fixed points is studied.