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Let M be a differentiable manifold and $f:M \to M$ be a diffeomorphism. Then $f$ induces a natural map $f^* :T^*M \to T^*M$. The pull back map $f^*$ is a symplectomorphism wrt the standard symplectic structure of the cotangent bundle $T^*M$. So it is a volum preserving map. Under what conditions is the mapping $f^*$ an ergodic map? How can one write the measure (associated to the volum form of $T^*M$) in the form of convex combination (or integral) of some ergodic measures? Are there some relations between these questions and some dynamical properties of $f:M \to M$?

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    $\begingroup$ Don't you need to assume there is compact subset of $T^*M$ invariant by $f^*$ and restrict the symplectic form to this subset in order to be able to have a finite measure space a properly define ergodicity? $\endgroup$
    – kvicente
    Commented Oct 2, 2023 at 15:22
  • $\begingroup$ @kvicente very interesting and necessary point. thank you I come back very soon $\endgroup$
    – Ali Taghavi
    Commented Oct 3, 2023 at 10:20
  • $\begingroup$ @kvicente what about isometric condition? namely we assume that f preserves a Riemannian metric However the volum form of the cotangent bundle is independent of any metric on M $\endgroup$
    – Ali Taghavi
    Commented Oct 3, 2023 at 10:50
  • $\begingroup$ @kvicente however i wonder if ergodicity can be defined in non finite measure too: a full measure set is a measurable set whose intersection with every compact set K has full measure $\mu(K)$ $\endgroup$
    – Ali Taghavi
    Commented Oct 3, 2023 at 10:53
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    $\begingroup$ It is possible to define ergodicity in non-compact spaces, there are easily findable references for that. However, it is a delicate matter. Now, because of the way symplectomorphisms of the cotangent bundle work, in order for $f^*$ to be ergodic a necessary condition would be that any $K\subset M$ with $\bar{K}\neq M$ and invariant by $f$, must have empty interior. So, this forces $K$ to be a rather complicate set, probably a Cantor like set where $f$ behaves somehow like a horseshoe map. $\endgroup$
    – kvicente
    Commented Oct 3, 2023 at 11:14

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