On cohomological \(C^0\)-(in)stability (Q380972)

Let \(C^0(M)\) be the space of continuous real functions on a compact metric space \(M\), endowed with the uniform norm. Given a homeomorphism \(f:M\to M\) , we define the space of \(C^0\)-coboundaries by \[ B(f,C^0(M)):=\{ v\circ f-v ; v\in C^0(M)\}. \] A homeomorphism \(f\) is said to be cohomologically \(C^0\)-stable iff \( B(f,C^0(M))\) is closed in \(C^0(M)\); see [\textit{A. Katok} and \textit{E. A. Robinson jun.}, Proc. Symp. Pure Math. 69, 107--173 (2001; Zbl 0994.37003)]. The author proves that a homeomorphism \(f\) is cohomologically \(C^0\)-stable if and only if it is periodic.