Generalizing results on dynamical systems on $\mathbb{R}^n$ to results on general manifolds

I am trying to self-learn dynamical systems but I am having the following problem: most books, especially introductory texts, all results are given as results about dynamical systems defined by evolution functions $f: \mathbb{R}^n \rightarrow \mathbb{R}^n $. To what degree can I expect these results to generalize to dynamical systems on manifolds given by $f: M \rightarrow TM$ and is there anything in particular I should be "careful" about.

Maybe to make the situation a little more specific here are some theorems I have thought about in particular:

1) Smales result that the (un)stable invariant manifold of a hyperbolic fixed point is an injective immersion of the (un)stable tangent space.

• Is there an analogy for manifolds? How would such a thing work on compact manifolds? It implies that there is some canonical way to map a subspace of the tangent space into a immersed submanifold, this should require extra structure on the manifold. Is it obvious where this comes from?

2) Shadowing lemmas

• For these clearly we need a metric on the manifold. Again, is there an obvious way to choose this? What if we have a hamiltonian system defined by a symplectic form, there is no "natural" way to define length, so how should I think about these things?

I would appreciate any help or perhaps even a recommended resource which will help me with these embarrassingly easy questions.

Solution 1:

The rule of thumb for results in ergodic theory is that everything local behaves just as it should for dynamical systems in $\mathbb R^n$- as can be seen by passing to charts or local trivializations of the tangent bundle- whereas global objects take more care to define.

1) Smales result that the (un)stable invariant manifold of a hyperbolic fixed point is an injective immersion of the (un)stable tangent space.

For a Riemannian manifold, the typical way to map tangent space onto the manifold is to use the exponential map, which maps a tangent vector $v \in T_x M$ to the solution to the time-one map for the geodesic equation with initial condition $(x, v)$. This map has the benefit of looking very close to isometric locally near the origin (corresponding to the point $x$) in $T_x M$.

Indeed, a typical practice when studying the local behavior along a given trajectory $\{ f^n x \}_{n \in \mathbb Z}$ for some $x \in M$ is to consider the connecting maps

$$ \tilde f_x = \exp_{f x}^{-1} \circ f \circ \exp_x $$ defined in a neighborhood of the origin in $T_x M$.

On the other hand, $\exp_x : T_x M \to M$ is never injective when $M$ is compact. This does not rule out that $\exp_x$ parametrizes a $W^u$ manifold through a point $x$ (see e.g. the unstable manifold through the origin for Arnold's Cat map on the torus), but it is certainly not always true either! These kinds of global considerations are typically more involved when working on a manifold.

2) Shadowing lemmas

Sometimes there is a 'canonical' choice of Riemannian metric given a symplectic form (check out almost complex structures). Otherwise, a quick-and-dirty way to do things is to pick a smooth Riemannian metric arbitrarily and just use that to define the shadowing property. This might not be very satisfying from the theoretical standpoint, but all smooth Riemannian metrics on a smooth compact manifold $M$ induce equivalent metrics (that is, equal up to a multiplicative constant)-- therefore, the shadowing property in one metric is equivalent to the shadowing property in another.