Definition.
Given two Poisson manifolds $M$ and $N$, a map $\phi:M\to N$ is a Poisson map if
$$ \{F,G\}_N=\{F\circ \phi,G\circ \phi\}_M. $$for all smooth functions on $N$.
$\blacksquare$
If the manifolds happen to be symplectic manifolds then $\phi$ is a symplectomorphism. In particular, if the manifolds are the phase space of a classical mechanical system then $\phi$ is a canonical map.
The Hamiltonian vector fields give rise to a local group of transformations which are Poisson maps (@olver86 proposition 6.16).
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Author of the notes: Antonio J. Pan-Collantes
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