We consider zero entropy $C^{\infty }$-diffeomorphisms on compact connected $C^{\infty }$-manifolds. We introduce the notion of polynomial growth of the derivative for such diffeomorphisms, and study it for diffeomorphisms which additionally preserve a smooth measure. We show that if a manifold M admits an ergodic diffeomorphism with polynomial growth of the derivative then there exists a smooth flow with no fixed point on M. Moreover, if dim M = 2, then necessarily M = ² and the diffeomorphism is $C^{\infty }$-conjugate to a skew...

Let L¹₄ be the group of 4-jets at zero of diffeomorphisms f of ℝ with f(0) = 0. Identifying jets with sequences of derivatives, we determine all subsemigroups of L¹₄ consisting of quadruples (x₁,f(x₁,x₄),g(x₁,x₄),x₄) ∈ (ℝ∖{0}) × ℝ³ with continuous functions f,g:(ℝ∖{0}) × ℝ → ℝ. This amounts to solving a set of functional equations.

The group of real analytic diffeomorphisms of a real analytic manifold is a rich group. It is dense in the group of smooth diffeomorphisms. Herman showed that for the $n$-dimensional torus, its identity component is a simple group. For $U\left(1\right)$ fibered manifolds, for manifolds admitting special semi-free $U\left(1\right)$ actions and for 2- or 3-dimensional manifolds with nontrivial $U\left(1\right)$ actions, we show that the identity component of the group of real analytic diffeomorphisms is a perfect group.