Let $f:M\to M$ be a partially hyperbolic diffeomorphism. That is, there exists a continuous splitting $TM=E^u\oplus E^c\oplus E^s$ into unstable, center and stable bundles. It is well known that there exist foliations $\mathcal{W}^u$ and $\mathcal{W}^s$ tangent to $E^u$ and $E^s$, respectively.


Let's assume $f$ is dynamically coherent. That is, there exist $\mathcal{W}^{cu}$ and $\mathcal{W}^{cs}$ tangent to $E^u\oplus E^c$ and $E^s\oplus E^c$, respectively, which are subfoliated by $\mathcal{W}^u$, $\mathcal{W}^s$ and $\mathcal{W}^{c}=\mathcal{W}^{cu}\cap \mathcal{W}^{cs}$.


Let's consider a center-stable manifold $W^{cs}(x)$.


Question: When does the following hold: $W^{cs}(x)=\bigcup\_{y\in W^c(x)}W^s(y)$? Is it always true, or do we need some extra assumption for it?


Another choice is to ask for $W^{cs}(x)=\bigcup\_{y\in W^s(x)}W^c(y)$. Just the previous one looks more natural. 



Center-stable manifolds
No, it is in general not true. An example is the one recently constructed by Rodriguez Hertz-Rodriguez Hertz-Ures. In that example (non-dynamically coherent in dimension 3) the union of stable manifolds through a center manifold is strictly contained in the center-stable manifold (its boundary is a strong stable manifold). You can see a drawing in [this slides](http://www.impa.br/opencms/pt/eventos/extra/2013_balzan/attach/apresentacao_maria_alejandra_hertz.pdf) of M.A.Rodriguez Hertz. Also, you can look at [these notes](http://www.cmat.edu.uy/~rpotrie/documentos/pdfs/minicurso082013.pdf) I wrote (page 21) were this possibility is also discussed. 


The question of wether such example can exist for a transitive partially hyperbolic diffeomorphism is open (see the paper by [Bonatti-Wilkinson](http://www.math.uchicago.edu/~wilkinso/papers/final.pdf) for more discussion). 


If your question was more devoted to understanding if the union of stable manifolds is contained in a center-stable one, see the paper of Burago-Ivanov in JMD (Proposition 3.1). For results in higher dimensions, see the papers by [Gogolev](http://www.math.binghamton.edu/agogolev/index_files/compact.pdf), [Carrasco](http://arxiv.org/abs/1105.0052) and [Bonhet](http://arxiv.org/abs/1303.4099). 


