Consecutive shifts along orbits of vector fields
  Let $M$ be a smooth ($C^{\infty}$) manifold, $F_1,...,F_n$ be vector fields on $M$ generating the corresponding flows $\Phi_1,...,\Phi_n$, and $\alpha_1,...,\alpha_{n}:M\to \mathbb{R}$ smooth functions. Define the following map $f:M\to M$ by $$f(x)= \Phi_n (... (\Phi_2 (\Phi_1 (x,\alpha_1(x)), \alpha_2(x)), ..., \alpha_n(x)).$$ In this note we give a necessary and sufficient condition on vector fields $F_1,...,F_n$ and smooth functions $\alpha_1,...,\alpha_{n}$ for $f$ to be a local diffeomorphism. It turns out that this condition is invariant with respect to the simultaneous permutation of the corresponding vector fields and functions. 
Sergey Maksymenko