Both Abraham-Marsden and Da Silva seem to imply that given a symplectomorphism $g:T^\ast X\to T^\ast X$ which preserves the tautological $1$-form $\alpha$, it must be that $g$ is fibre preserving. 


In fact, this is addressed in the question
[Cotangent bundle lift theorem](https://mathoverflow.net/questions/151083/cotangent-bundle-lift-theorem).


The answer given to this question stresses very clearly that g must preserve fibres to have any chance of being a lift, even if it preserves the canonical 1-form. 


I would really like to see a counterexample to this. That is, a symplectomorphism which preserves the canonical $1$-form but is not the lift of a diffeomorphism. 


I have struggled with trying to prove both Abraham-Marsden and Da Silva's claim. Any help would be greatly appreciated. 


I can show the following:


If $V$ is the symplectic dual of $\alpha$ (i.e. $\omega(V,\cdot)=\alpha$), then $g\_\ast V=V$, or equivalently $g\circ\theta\_t=\theta\_t\circ g$ where $\theta\_t$ is the flow of $V$. Also I can show that $\theta\_t$ is fibre preserving (i.e. $\theta\_t(T\_p^\ast X)=T\_p^\ast X)$. This last fact follows from actually computing $\theta\_t$ which is $\theta\_t(p,\xi\_p)=(p,e^t\xi\_p)$.



Lifting a Diffeomorphism to the Cotangent Bundle
The vector field $V$ is the radial vector field in the fibers, and it vanishes along the zero locus of $\alpha$, which is the zero section $Z\subset T^\*M$. Thus, $g$ must preserve $Z$ and hence be a diffeomorphism of $Z$, say $f:Z\to Z$. Since $Z\to M$ is a diffeomorphism, this can be regarded as a diffeomorphism $f:M\to M$. Since $g$ preserves $V$, it must, for each $z\in Z$, i.e., $z = 0\_x$ for $x\in M$, carry the $\alpha$-limit set $S\_z$ of $z$ with respect to $V$ into (and diffeomorphically onto) the $\alpha$-limit set of $g(z) = 0\_{f(x)}$, but $S\_{0\_x}=T^\*\_xM$ and $S\_{0\_{f(x)}}=T^\*\_{f(x)}M$. Thus, $g(T^\*\_xM) = T^\*\_{f(x)}M$. The rest is now clear, I think. 


