Chaos and Homology Theory

will discuss, the existence of Morse-Smale diffoemorphisms and relations with the entropy conjecture. Smooth diffeomorphosms without chaotic behaviour must induce a map in homology whose eigenvalues are roots of unity (and more). That is if some eigenvalue is not a root of unity there must be chaotic behavior. These are old results in which I have recently become interested again, there are quite a few interesting open problems.