Topological classification of linear endomorphisms

Letf=f ₊f _–f f ₀ be a linear direct sum decomposition of a real endomorphismf End ⁿ , where the components correspond respectively to absolute values of eigenvalues ||: 0<||<1(f ₊); ||>1(f _–); ||=0(f ); and ||=1(f ₀). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off ₊ and off _–, together with the linear isomorphy classes off andf ₀. We prove that this is true in case ⁿ contains nof-periodic points of periods 5 or 7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.Research supported by the National Science Foundation (Contract number 144-B 695), the Wisconsin Alumni Research Foundation (project number 120432), and l'Institut des Hautes Etudes Scientifiques.