Geometrical and maximal u-entropy measure-theoretic structures of maps with c-mostly expanding center and c-mostly contracting

Abstract: In this paper, we classified ergodic maximal $u$-entropy measures among the set of partially hyperbolic diffeomorphisms with $(E^{uu},A)$-Markov partition, $c$-mostly contracting center and $c$-mostly expanding center in the $C^1$-topology. More specifically, ergodic maximal $u$-entropy measures can be divided into $c$-$u$-states and $c$-$cu$-states. Due to the existence of $c$-mostly expanding center, ergodic maximal $u$-entropy measures of type $c$-$u$-states are trivial. For ergodic maximal $u$-entropy measures of type $c$-$cu$-states, by noting that its structure is similar to that of physical measures, with the help of the skeleton we obtain their properties of supports and basins, and variation of number in some sense. Consequently, the ergodic measures of maximal $u$-entropy of type $c$-$cu$-states are finite in the sense that the measure supports are consistent. On the certain environment, union of the basins of ergodic measures of maximal $u$-entropy of type $c$-$cu$-states is dense in whole space and number of $c$-$cu$-state is upper semicontinuous in the sense of support. Finally, we provide propositions for qualitative construction examples, which also explains that there are many such examples here.