# 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.

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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: