It it possible to have a flow $\phi(t,x)$ on a 2-dimensional manifold where for some $t > 0$, the map
$g(x) := \phi(t,x)$
creates a horseshoe?
By $\phi(t,x)$ I mean the solution to the ODE defining the flow on the manifold at time $t$, from starting time $0$ and starting position $x$.
There seem to be a number of iterated maps, like the toral automorphisms, or of course the standard horseshoe map, which create horseshoes. But they don't seem like they can be realized as time-1 maps of flows, and I'm wondering if there's a general fact that this can't be done. Basically any references or insight about when a map can be realized as a flow in general would be really helpful as well.
Thanks.
Your $g(x)$ is strangely defined (as the expression on the right-hand side is $t$-dependent) but I think I get the problem. So imagine you have a smale horsehoe generated by a 2 dimensional discrete dynamical system (by which I mean you have the unstable manifold intersecting the stable manifold in such a way that you get a smale horseshoe). How do you see this back in an ODE. You actually cannot have this in a 2 dimensional ODE you really need a third dimension since solution curves cannot self-intersect. To view this in 3 dimensions imagine that the 2-dimensional discrete dynamical systems corresponds to the poincare map associated to the 3 dimensional ODE. If anything needs more explanation let met know.