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

Abstract

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 \( C^{\infty} \) 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}.