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I do not understand how this shows 'no, that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $p(T) = d_\gamma(0) \phi_T ^* p(0)$ the latter equation being the necessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a 'yes' example. Take $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of $p$, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]

I do not understand how this shows 'no, that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $(D_{\gamma(0)}\phi_T )^* p(T) = p(0)$ the latter equation being the necessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a 'yes' example. Take $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of $p$, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]

I do not understand how this shows no, that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $p(T) = d_\gamma(0) \phi_T ^* p(0)$ the latter eqn being the necessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$  ! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a `yes' example. Take  $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of p, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]

I do not understand how this shows 'no, that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $p(T) = d_\gamma(0) \phi_T ^* p(0)$ the latter equation being the necessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a 'yes' example. Take $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of $p$, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]

I do not understand how this shows no', that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no'. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $p(T) = d_\gamma(0) \phi_T ^* p(0)$ the latter eqn being the neccessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$ ! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a yes' example. Take $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of p, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]

I do not understand how this shows no, that no such flow exists'. But here is a tweak on this thinking that does yield a proof of no. Suppose $(\gamma (t), p(t))$ is periodic orbit upstairs with period $T >0$. Then $\gamma (t) = (x(t), y(t))$ is periodic downstairs with period $T$ and $p(T) = d_\gamma(0) \phi_T ^* p(0)$ the latter eqn being the necessary and sufficient condition that the lift of a periodic orbit downstairs of period $T$ be periodic upstairs. This condition is a linear condition on $p$ ! Hence for all $\lambda$ real the solutions with initial condition $(\gamma(0), \lambda p(0))$ are also periodic of period $T$: there are infinitely many if there is one. [By either projectivizing the fiber of the cotangent bundle or by fixing energy you can get a `yes' example. Take $\phi_t$ to have one hyperbolic limit cycle surrounding a single hyperbolic fixed point. Then the only periodic solution upstairs, modulo scaling of p, is the one projecting onto the limit cycle and having 0 momentum normal to the limit cycle.]