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Let $\mathcal{F}$ be a Foliation on a Manifold $M$ and $g:M\longrightarrow M$ a Diffeomorphism. We say that the foliation $\mathcal{F}$ in invariant under the diffeomorphism $g$ if the diffeomorphism $g$ takes leaves to leaves (i.e., if $F$ is a leaf of the foliation $\mathcal{F}$ then so is $g(F)$).

The question is:

Is this definition equivalent to the condition $g^{\star}(\mathcal{F})=\mathcal{F}$, where $g^{\star}(\mathcal{F})$ is the Pullback Foliation of $\mathcal{F}$ by $g$.

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If $g\colon M\to M$ is a diffeomorphism and $A$, $B$ are any subset of $M$, then $g(A) = B \iff A = g^{-1}(B)$.

By definition, the leaves of the pullback foliation $g^{-1}\left(\mathcal{F}\right)$ are the connected components of $g^{-1}(F)$ where $F$ runs over the leaves of $\mathcal{F}$. As diffeomorphisms are homeomorphisms, they preverve connectedness, and the answer to your question is: yes.

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