Diffeomorphisms: Denjoy Theory

The topological theory of diffeomorphisms of the circle was started by A. Denjoy, almost fifty years after H. Poincaré introduced the concept of rotation number. In a seminal paper published in 1932, Denjoy (J. Math. Pure et Appl 11:333–375, 1932) proved that every sufficiently smooth circle diffeomorphism f without periodic points is topologically equivalent to an irrational rotation. Here, the expression “sufficiently smooth” means that f is \(C^1\) and \(\log {Df}\) is a function of bounded variation.