Can orientation preserving diffeomorphism in $\mathbb{R}^d$ be presented by flowmap of dynamical systems?

Question

Because flowmaps are homeomorphic maps, I was wondering if there is any literature that proves that diffeomorphism $\Phi(x)$ can be expressed as a flowmap of a certain dynamical system?

that is, does there exist $\dot z(t)=v(z(t))$ with initial value $z(0)=x$ such that $\Phi(x)=\phi_v^\tau(x)$ where $\phi_v^\tau(x)=z(\tau)$?

Answer 1

No, this is usually not possible. There's a previous MO question discussing this, but to add to the material there:

• A time-one map $\phi_1$ commutes with the 1-parameter family of diffeomorphisms $\phi_t$. However, The C1 generic diffeomorphism has trivial centralizer (Bonatti, Crovisier, Wilkinson). So being a time-one map is a rare property, in a sense!
• One very simple obstruction can arise if $\Phi$ has periodic points. For a flow map, a point with period greater than one must correspond to a closed orbit, so the derivative must have at least one eigenvalue of 1. This, too, is a rare property.
• Another set of concrete examples: consider diffeomorphisms of $\mathbb{R}^2$, which can have complicated chaotic dynamics. For flows in $\mathbb{R}^2$, you can't have chaos: recurrence is constrained to be relatively simple, by the Poincare-Bendixson theorem.