# A local transitivity property of the automorphism group of a foliated manifold

Let $$(M,\mathcal F)$$ be a smooth foliated manifold. An automorphism of $$(M,\mathcal F)$$ is a diffeomorphism of $$M$$ that takes leaves of $$\mathcal F$$ onto leaves. Let now $$L$$ be a leaf of $$\mathcal F$$. It may happen that $$L$$ has an open neighborhood $$U$$ which is a sum of leaves and such that for every leaf $$L'\subset U$$ there exists an automorphism $$\phi$$ taking $$L$$ onto $$L'$$.

My questions are:

Does every $$(M,\mathcal F)$$ have a leaf $$L$$ with the described property?

If the answer is no, how much can we hope for instead? And is there a simple counterexample? Are there some natural classes of foliations which still have this property?

Context:
Ideally, I would like to consider a smooth foliated manifold $$(M,\mathcal F)$$ such that in some flat chart $$\psi\colon U\to \mathbb R^n$$ there exists an open set $$V\subset U$$ and a family $$\tau_x$$ of automorphisms of $$(M,\mathcal F)$$ indexed by a neighborhood of 0 in $$\mathbb R^n$$, where $$\tau_x$$ acts on $$V$$ like a translation by $$x$$ in the chart $$\psi$$. This condition is satisfied for instance when there is a neighborhood $$U$$ of a leaf $$L$$ and a foliation-preserving diffeomorphism $$U\to L\times W$$, where $$L\times W$$ has the corresponding product foliation.

No. There exist foliations $F$ of the plane that are rigid in the sense that no automorphism of $F$ can permute the leaves. See the following paper: