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Let $M$ be a closed compact Riemannian manifold, $\mathcal{F}$ be a $C^1$ foliation on $M$. Let $F(x)\in\mathcal{F}$ be the leaf containing $x$.

Definition. $\mathcal{F}$ is said to be a unstable foliation, if there exist $\lambda>1$, and a diffeomorphism $g:M\to M$ such that
- preserves $\mathcal{F}$: $\quad$ $g(F(x))=F(gx)$, for all $x\in M$.
-- expands $\mathcal{F}$: $\quad$ $\|Dg(v)\|\ge \lambda$, for all unit vector $v\in T_yF(x)$, $y\in F(x)$ and $x\in M$.

The foliation $\mathcal{F}$ may coincide with the unstable foliation of $g$ under some extra assumption (for example, a dominated splitting).

Definition. $\mathcal{F}$ is said to be
- transitive, if $\overline{F(x)}=M$ for some $x\in M$;
- minimal, if $\overline{F(x)}=M$ for all $x\in M$.

There are transitive foliaitons without being minimal. For example, the geodesic flow $\phi_t:T^1S_g\to T^1S_g$, on a compact surface with constant negative curvature. The flow-line foliation is transitive, but not minimal. These foliations can't be unstable since there are closed leaves (or, closed orbits).

Question 1. Let $\mathcal{F}$ be a transitive unstable foliation. When will it be minimal?

Question 1 may be too general to consider. What about the following special case?

Question 2. Suppose $\dim M=3$ and $\dim F(x)=1$. If $\mathcal{F}$ is transitive unstable foliation, when will it be minimal?

There are some characterization of transitivity and minimality by the $C^\ast$-algebra of $\mathcal{F}$. I don't know if that can help.

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Recently, Y. Shi has constructed a partially hyperbolic diffeomorphism in a nilmanifold where the unstable foliation is not minimal (the global dynamics is Axiom A, so it has a hyperbolic attractor). The example is not yet written, but some insight can be gained from this paper of Bonatti-Guelman.

In our paper with A. Hammerlindl we show that every partially hyperbolic diffeomorphism of a (non-toral) nilmanifold has "transitive unstable foliation" (i.e. has a dense unstable leaf).

The general question you ask is too broad, but this at least shows that there are examples where transitivity does not imply minimality.

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  • $\begingroup$ Rafael, probably non-trivial nilmanifold? $\endgroup$ – Andrey Gogolev Aug 16 '13 at 16:32
  • $\begingroup$ Thank you Rafael! Another viewpoint is that minimal unstable foliation implies the diffeo $g$ must be transitive (mixing). So it suffices to destroy the transitivity of $g$. $\endgroup$ – Pengfei Aug 16 '13 at 20:34
  • $\begingroup$ So combining your result with Shi's, even Q2 is false. $\endgroup$ – Pengfei Aug 16 '13 at 20:50
  • $\begingroup$ Andrey, yes, I corrected that. Pengfei, yes, Q2 is "false" too thanks to that (quite unexpected, at least to me) example.- $\endgroup$ – rpotrie Sep 6 '13 at 1:00
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This is an interesting question. Little is known. First of all if g is Anosov then transitivity implies miniamlity.

More generally, there are various assumptions that guarantee minimality. See for example "A sufficient condition for robustly minimal foliations" by Pujals and Sabmarino.

In my "Smooth conjugacy of Anosov diffeomorphisms on higher-dimensional tori" there are examples of transitive foliations which are not known to be minimal. It would be interesting to do computational work to see what the answer should be.

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  • $\begingroup$ Hi Andrey! For transitive Anosov, the unstable foliations are minimal. Is this what you mean? Or do you mean, for general Anosov, if $\mathcal{W}^u$ is transitive, $\mathcal{W}^u$ is also minimal? $\endgroup$ – Pengfei Aug 14 '13 at 22:12
  • $\begingroup$ Pengfei, if W^u is transitive then the diffeomorphism has to be transitive (by spectral decomposition) and then one argues that W^u is minimal. I believe same is true for Anosov flows (or time-1 maps if you prefer diffeos): if W^u is transitive, then g is not suspension and is transitive, then one argues that W^u is in fact minimal. I think this is carefully written in one of Plante's classical papers. $\endgroup$ – Andrey Gogolev Aug 14 '13 at 23:43
  • $\begingroup$ Sorry I said something nonsensical. As you say: if g is transitive then W^u is minimal. If g is not transitive then W^u is not minimal, by spectral decomposition. However, it seems that W^u can be (potentially) transitive when g is not. $\endgroup$ – Andrey Gogolev Aug 15 '13 at 0:13
  • $\begingroup$ I had the Anosov problem for a while. Even 2D case gives me a headache. We know all 2D Anosov diffeo are transitive and hence $W^u$ is minimal. But I can't derive the minimality directly from transitivity of $W^u$ on $\mathbb{T}^2$. $\endgroup$ – Pengfei Aug 16 '13 at 1:42
  • $\begingroup$ I don't see how to do it either, just from transitivity of W^u. You also need to use that periodic points are dense. $\endgroup$ – Andrey Gogolev Aug 16 '13 at 13:09

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