Minimal sets of certain annular homeomorphisms (Q1860645)

The authors deal with a homeomophism of the annulus \(S^1\times \mathbb{R}\) of the form \[ F_{\alpha,\varphi}(x,y)=(x+\alpha,y+\varphi(x)),\tag{1} \] where \(\alpha\) is an irrational number and \(\varphi\) is a continuous function on \(S^1\) with vanishing integral. The authors show that, if \(\varphi\) is of bounded variation and if \(F_{\alpha,\varphi}\) is not topologically conjugate to \(F_{0,\varphi}\), then \(F_{\alpha,\varphi}\) does not admit a minimal set. Next the authors show the abundance of homeomorphisms without minimal set.