Let $X$ be a tangent vector field on a compact manifold $M$ and suppose there is an open set $U \subset M$ with the following property:
There exists a $T>0$ such that $\displaystyle \bigcap_{|t|<T} X_t(M) \subset U$
(where $X_t$ indicates the flow of $X$ at time $t$)
Does the property hold for the same open set $U$ if one changes the field $X$ by any field within a close neighbourhood of $X$?
By close neighbourhood i mean all fields $Y$ such that the values of $Y$ and its derivatives up to $k$-th order are $\epsilon$-close to those of $X$ on all $M$
Also for context, here's where i was reading from. This is from chapter 4 of "Geometrical theory of dynamical systems" (Palis & de Melo)
From the text i guess the result is true, but it doesn't seem obvious to me
