Limit sets of typical homeomorphisms (Q2888774)

The authors study properties of limit sets \(\omega(f,x)\) of a typical homeomorphism \(f: X \to X\). For \(f: X \to X\), the limit set \(\omega(f,x)\) of \(f\) at \(x\) is defined as the set of all limit points of the sequence \((f^j(x))_{j \geq 0})\). The authors prove the following result:NEWLINENEWLINE Given an integer \(n \geq 3\), a mertizable compact topological \(n\)-manifold \(X\) with boundary and a finite positive Borel measure \(\mu\) on \(X\), then for the typical homeomorphism \(f:X \to X\), it holds that for \(\mu\)-almost every point \(x\) in \(X\) the limit set \(\omega(f,x)\) is a Cantor set of Hausdorff dimension zero, each point of \(\omega(f,x)\) has a dense orbit in \(\omega(f,x)\), \(f\) is non-sensitive at each point of \(\omega(f,x)\), and the function \(a \to \omega(f,a)\) is continuous at \(x\).