Given a compact manifold $M$ and a vector field as a map $F: M \rightarrow TM$. We can define a Dynamical System (DS) on $M$ by setting $$\dot{x}=F(x).$$
There are some dynamical properties of this DS that are induced by topology. I.e. the Poincare-Hopf-theorem states that the index of the vector field is dictated by the Euler characteristic of the manifold
$$\sum_{i} \operatorname{index}_{x_{i}}(v)=\chi(M)$$
E.g., in practice, this means that a system with two stable FPs in $\mathbb{R}^2$ must have a saddle (or some more complicated structure like a line attractor) between the two stable FPs.
My question is the following: Let's go to multistable $3$-dimensional (or $n$-dimensional) systems with multiple chaotic attractors. Are there still theorems like the Poincare-Hopf theorem? I.e. given a specific chaotic attractor $A_1$ can we topologically infer that there has to exist another chaotic attractor $A_2$?
I am specifically thinking about situations where the basins of attraction of two attractors are fractal. My intuition would tell me that the fractal structure of one of the basins might induce the fractal structure of the other basin.