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

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

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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![Blockquote using this topology he proves the following theorem:enter image description here the hypothesis of Theorem $A$ are: enter image description here


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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 $\mathfrak{B}$ enter image description here enter image description here

Can anyone help me?


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