Friday, 29 May 2015

symplectic topology - Anosov diffeomorphisms and the chaotic hypothesis

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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