Topological transitivity for a class of monotonic mod one transformations

Abstract

Suppose that f : [0, 1] → [0, 2] is a continuous strictly increasing piecewise differentiable function, and define Tfx := f(x) (mod 1). Let \({\beta \geq \sqrt[3]{2}}\). It is proved that Tf is topologically transitive if inf f′ ≥ β and \({f(0)\geq\frac{1}{\beta+1}}\). Counterexamples are provided if the assumptions are not satisfied. For \({\sqrt[3]{2}\leq\beta 2 -\beta-\frac{1}{\beta^2+\beta}}\).