What meaning of this sentence?

Question

I study about Poisson geometry. You should be know that every Poisson structure induced singular foliation. I encountered this sentence

“Two points lie in the same leaf if and only if one is accessible from the other through a composition of local homiltonian flows”

What’s meaning of right part of proposition?

Answer 1

This is explained e.g. in section 1.3.4 Global Structure: The Symplectic Foliation of the book Poisson Structures by Laurent-Gengoux, Pichereau, and Vanhaecke:

By a Hamiltonian path from $m$ to $m'$ we mean a curve $\gamma$, defined on an open neighborhood of $[0,1]$, with $\gamma(0) = m$ and $\gamma(1) = m'$, which is an integral curve of a Hamiltonian vector field $\mathscr{X}_F$, where $F$ is a function, defined on an open neighborhood of $\gamma([0,1])$. More generally, when points $m_0, \ldots, m_N$ in $M$ are such that there exists a Hamiltonian path from $m_{i-1}$ to $m_i$ for $i=1,\ldots,N$, we say that there exists a piecewise Hamiltonian path from $m_0$ to $m_N$.

The section continues to build up to the

Theorem 1.30. Every Poisson manifold $(M,\pi)$ is the disjoint union of immersed submanifolds, whose tangent spaces are spanned by the Hamiltonian vector fields of $(M,\pi)$. The Poisson structure, restricted to each of these submanifolds yields a Poisson structure of maximal rank (a symplectic structure). This decomposition is called the symplectic foliation of $M$ and the immersed submanifolds are called the symplectic leaves of $M$. For $m \in M$ the symplectic leaf which contains $m$ is given by $$\mathscr{I}_m(M) = \{m'\in M \mid \exists \text{ a piecewise Hamiltonian path in $M$ from $m$ to $m'$} \}.$$

and they remark

The first part of the theorem can also be restated by saying that the (singular) distribution, defined by the Hamiltonian vector fields, is integrable (admits an integral manifold through each point).