A note about topologically transitive cylindrical cascades (Q5957367)

The main result of the paper is the following. Assume that \(\sigma:\Sigma\rightarrow \Sigma\) is a topologically mixing subshift of finite type, that \(\beta:\Sigma\rightarrow R\) is continuous, and consider the skew product \(\sigma_\beta:\Sigma\times R\rightarrow \Sigma \times R\) defined by \(\sigma_\beta(x,t) = (\sigma(x),t + \beta(x))\). Assume that there is a point \((x,t)\) such that the sequence of second coordinates of \(\sigma_\beta^n(x,t)\), \(n\geq 0\), is unbounded above and below. Then there are arbitrarily small \(C^0\) perturbations \(\gamma\) of \(\beta\) such that \(\gamma\) is Hölder and the corresponding skew product \(\sigma_\gamma\) is topologically transitive. A simple example is described showing that the hypotheses above do not imply that \(\sigma_\beta\) is topologically transitive. Also, it is observed that the result of the theorem fails if the hypothesis is weakened from assuming that the real coordinates of a given semi-orbit are unbounded in both directions to assuming that the real coordinates of some full orbit are unbounded in both directions.