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.