Take the standard map with its break down of invariant circles and all,
Cantorii and the like. How would you start to approximate by Anosov?
I am thinking of Aubry/Mather/Bangert results on 2 degree of freedom
Hamiltonian systems, or, area preserving maps of an annulus to itself.
There are various theorems to the effect that these systems are a kind
of unentanglable mess of integrable and ``chaotic''.
On the flip side, there are results of Gole/Boyland asserting that IF
a natural mechanical system: kinetic + potential, admit a hyperbolic metric:
so the system is on $T^*Q$, and $Q$ admits a hyperbolic metric, then the system
is ``semi-conjugate'' to the corresponding Anosov system.
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