# Structural Stability on Compact $2$-Manifolds with Boundary

I'm studying the structural stability of vector fields and I'm interested in learning about this phenomenon on compact $$2$$-manifolds with boundary.

Let $$M^2$$ be a compact connected 2-manifold and $$\mathfrak{B}$$ the space of vector fields with the $$\mathcal{C}^1$$ topology. The paper "M. M. Peixoto - Structural stability on two-dimensional manifolds" defines structural stability as

Using this definition Peixoto was able to demonstrate the following theorem:

My Question: I would like to know if someone knows a result, like the above, about Structural Stability in a compact $$2$$-manifold with boundary (the case when $$M^2 \subset \mathbb{T}^2$$ is enough for my purposes).

## Just a few comments about my search for the result

Searching online I found the paper Clark Robison - Structural stability on manifolds with boundary, however, the author defines a strange topology using the flows of the vector field, and I don't know if it is the same topology as in definition 1 or if it implies the result that I want. Moreover, the demonstration is a bit confusing and I think that important points are missing, e.g. the demonstration of Theorem C (in my humble opinion). Besides I don't even know if this would imply that the set of Vector Fields that are structurally stable would be at least dense in the set $$\mathfrak{B}.$$

His approach is using the $$\mathcal{C}^1$$-topology in the flow space, and not the $$\mathcal{C}^1$$-topology of the vector fields using this topology he proves the following theorem: the hypothesis of Theorem $$A$$ are:

where

It is also worth to mention that on the paper "M.C. Peixoto and M.M. Peixoto - Structural Stability in the Plane with Enlarged Boundary Condition", the following result was proved. Let $$G\subset \mathbb{R}^2$$ be a compact region, such that the boundary $$L$$ of $$G$$ is a $$\mathcal{C}^1$$ simple curve, then $$X$$ (a vector field in $$G$$) is structurally stable $$\Leftrightarrow$$ $$X$$ satisfies conditions $$A$$ and $$B$$, it has also been proven that the vector fields that this property possesses are open and dense $$\mathfrak{B}$$

Can anyone help me?