# When is a foliation invariant under a diffeomorphism?

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 $$g$$.

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.