Twisted Rokhlin property for mapping class groups

In this paper, generalising the idea of the Rokhlin property, we explore the concept of the twisted Rokhlin property of topological groups. A topological group is said to exhibit the twisted Rokhlin property if, for each automorphism $\phi$ of the group, there exists a $\phi$-twisted conjugacy class that is dense in the group. We provide a complete classification of connected orientable infinite-type surfaces without boundaries whose mapping class groups possess the twisted Rokhlin property. Additionally, we prove that the mapping class groups of the remaining surfaces do not admit any dense $\phi$-twisted conjugacy class for any automorphism $\phi$. This supplements the recent work of Lanier and Vlamis on the Rokhlin property of big mapping class groups. We also prove that the mapping class group of each connected orientable infinite-type surface without boundary possesses the $R_\infty$-property.