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.



Unstable Foliations
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](https://www.aimsciences.org/journals/displayArticles.jsp?paperID=5161) of Bonatti-Guelman. 


In [our paper](http://arxiv.org/abs/1302.0543) 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. 


