Dynamics of the radix expansion map (Q1321863)

Let \(\phi(x) = (\beta x + \alpha)\pmod 1\) be a map of \([0,1]\) onto itself and \(1 < \beta < 2\), \(\alpha \geq 0\). It is shown that \(\phi\) is chaotic and the number of its periodic points of period \(n\) is \(O(\beta^ n)\). The former means that \(\phi\) is topologically transitive, its periodic points are dense and it is sensitive to initial conditions.