We provide two robust examples of globally partially hyperbolic systems with a multi one-dimensional center splitting, for which all Gibbs u-states are hyperbolic and the number of physical measures is fixed. In the second example, the physical measures have different unstable indices, and the supports of those with index at least 2 can be adjusted to lie on a non-uniformly hyperbolic set using the techniques developed for the first example. Such partially hyperbolic systems were previously studied in MC25, Bur25, where the existence and finiteness of physical measures were established under the assumption of hyperbolicity of the Gibbs u-states or the SRB measures, respectively.
Mi et al. (Fri,) studied this question.