Transverse minimal foliations on unit tangent bundles and applications

Abstract

We show that if $\mathcal{F}_1$ and $\mathcal{F}_2$ are two transverse minimal foliations on $M = T^1S$ then either they intersect in an Anosov foliation or there exists a Reeb-surface in the intersection foliation. The existence of a Reeb surface is incompatible with partially hyperbolic foliations so we deduce from this that certain partially hyperbolic diffeomorphisms in unit tangent bundles are collapsed Anosov flows. We also conclude that every volume preserving partially hyperbolic diffeomorphism of a unit tangent bundle is ergodic.

Figures (20)

• Figure 1: A Reeb surface and its lift to the universal cover.
• Figure 2: The set $D_\varepsilon(I)$ for $I = [\xi,\eta]$ is the complement of the union of $H_+(\ell)$ and $B_C(\ell)$. If $\varepsilon$ is very small, than $C$ is very large.
• Figure 3: The limit set of $r_1$ is the interval $I_1$ with endpoints $a, b$. $g_1, g_2$ are geodesic rays with ideal points in the interior of $I_1$. The segments $v_n$ are segments in $r_1$ which limit to $[a,b]$. The points $z_k$ are in $r_2$ and have to connect outside the $v_n$ to one of $g_1$ or $g_2$ in $D_{n_k}$. So the intersections with (say) $g_1$ limit to an arbitrary point in the interior of $I_1$.
• Figure 4: An example where the negative ray of $c$ lands and the positive ray of $c$ accumulates in an interval which is not a point.
• Figure 5: Intersection in more than one connected component forces non-Hausdorff leaf space of the induced one dimensional foliation.

Theorems & Definitions (124)

• Theorem 2.1: Matsumoto Matsumoto
• Corollary 2.2
• Proposition 2.3
• proof
• Remark 2.4
• Lemma 2.5
• proof
• Proposition 2.6
• proof
• Corollary 2.7