The topological Rohlin property and topological entropy (Q2772969)

Let \(X\) be a compact metric space and \(G\) denote the group of homeomorphisms \(X\to X\) with the topology of uniform convergence. \(X\) satisfies the topological Rohlin property if the action of \(G\) on itself by conjugation has dense orbits. It is shown that the Hilbert cube and Cantor sets satisfy this property. Further, these results enable to show that zero entropy is generic for homeomorphisms of Cantor sets while infinite entropy is generic for homeomorphisms of the Hilbert cube or cubes of dimension \( d\geq 2\).