Chaotic and topological properties of β-transformations

In this paper, we prove that the β-transformations are chaotic in the sense of both Li–Yorke and Devaney. The topological and metric properties of the sets of points with dense or non-dense orbits are investigated. We also prove the result that the set of points with non-dense orbits under the β-transformation is of full Hausdorff dimension for any β>1.