$C^k$ conjugacy of 1-d diffeomorphisms with periodic points

Abstract:

It is shown that the set of heteroclinic orbits between two periodic orbits of saddle-node type induces functional moduli which are completely contained in a new ‘transition map’. For one-dimensional $C^2$ diffeomorphisms with saddle-node periodic points, two such diffeomorphisms are $C^2$ conjugated if and only if the transition maps of their heteroclinic orbits are the same. An equivalent transition map is given for $C^k$ diffeomorphisms with hyperbolic periodic points, and it is shown that this transition map is an invariant of $C^k$ conjugation. However, in this case the transition map alone is sufficient to guarantee conjugacy only in a limited sense.