Stable and unstable branched surface

Description

A stable branched surface is a surface that branches in the direction of a Dynamic vector field

Definition

Let $M$ be a three-manifold and $X$ a Dynamic vector field. Suppose that $B^{\ast }|subseteqM$ is a properly embeddedBranched surface. We call $B^{\ast }$ a stable branched surface with respect to $X$ if it fulfils the following:

• $X$ is nowhere perpendicular to a Sector (branched manifold) of $X$ This naturally induces a vectorfield $X^{\ast }$ on $B$ by projection, called the upward semi-flow

• The semi-flow $X^{\ast }$ is transverse to the branch locus of $B^{\ast }$. It points from the side with fewer sheets to the side with more (we assume there are always differently many sheets)

• The semi-flow $X^{\ast }$ is never orthogonal to the branch locus. (Why?)

  For the definition of the unnstable branched surface, the second requirement is reversed.

Mosher uses a slightly different definition.

Properties

Examples

Upper and lower branched surfaces

Let $t$ be a Transverse veering tetrahedron. Let $X_t$ be a non-vanishing Vector field with the following properties

• The vector field $X_t$ is orthogonal to each face of $t$
• Each orbit of $X_t$ connects a lower face $t$ with an upper face
• The branched surfaces $B^t$ and $B_t$ (in dual position) are stable and unstable with respect to $X_t$

We define $X_{\mathcal{V}}$ by gluing together the vector fields $X_t$ for all $t$. Then the Upper and lower branched surface $B^{\mathcal{V}}$ and $B_{\mathcal{V}}$ are stable and unstable with respect to $X_{\mathcal{V}}$.