Finiteness and Geometric structure of $c$-$cu$-States with Maximal $u$-Entropy

In a $c$-mixed system, we study $c$-$cu$-states, which capture the structural characteristics of physical measures (in similar systems), having maximum $u$-entropy. It is shown that the maximum number of $c$-$cu$-states with pairwise distinct supports is finite, and Proposition~\ref{pro.con} is provided to construct such systems. Using a modified version of Smale’s method \cite{Smale}, we explicitly construct a $C^{\mathrm{\infty }}$ diffeomorphism $f$ on ${\mathbb{T}}^4$ with a partially hyperbolic splitting: $$F^{uu}{\oplus }_{\succ }{F}^{cu}{\oplus }_{\succ }(F^{cs}{\oplus }_{\succ }{F}^{ss}),$$ such that $f$ has a mixed center (or $c$-mixed center), $F^{cu}$ is not uniformly expanding, and $F^{cs}\oplus F^{ss}$ is not uniformly contracting. The method can be used to modify the product maps of linear Anosov skew products and linear Anosov systems, such that the modified map has a mixed center (or $c$-mixed center) and is a skew product of linear Anosov skew product. This provides concrete examples to illustrate how the physical measure changes in a semicontinuous manner across the system when the corresponding $E^{cu}$ is non-uniformly expanding and the corresponding $E^{cs}$ is non-uniformly contracting. The study of physical measures in similar systems can be found in the literature \cite{ref7, CM}.