Let $f:M\rightarrow M$ be a diffeomorphism and $\Lambda$ be a hyperbolic set, where $M$ is a compact Riemannian manifold without boundary. In the definition of the hyperbolic set $\Lambda$, the tangent space over $\Lambda$ splits into two subbundles $T_{x}M=E_x^s\bigoplus E_x^u$ for all $x\in\Lambda$. Then there are conclusions that the dimensions of $E_x^s$ and $E_x^u$ are locally constant and those subspaces change conituously.
Thus, is it possible that the dimension of $E_x^s$(or $E_x^u$) changes as $x\in\Lambda$ varies ? Thanks for any help.
Indeed, in general the dimension may vary but it is locally constant, that is, in any arbitrarily small neighborhood of a given point in the hyperbolic set the dimensions of the stable and unstable spaces are constant in that neighborhood (precisely due to continuity, since the dimensions are integers). The compactness of $M$ does not interfere in any way.
As a consequence, you may simply take two fixed hyperbolic fixed points $p$ and $q$ of different stable (and unstable) dimension, and their union $\{p,q\}$ will be a hyperbolic set for which the stable and unstable spaces are not equal at every point.