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. © 2011 Elsevier Inc.
Li, B., & Chen, Y. C. (2011). Chaotic and topological properties of β-transformations. Journal of Mathematical Analysis and Applications, 383(2), 585–596. https://doi.org/10.1016/j.jmaa.2011.05.049