Uniform recurrence properties for beta-transformation

作者：Zheng, Lixuan^*; Wu, Min

来源：Nonlinearity, 2020, 33(9): 4590-4612.

DOI：10.1088/1361-6544/ab8a65

• beta-transformation
• Hausdorff dimension
• uniform recurrence property

For any beta> 1, let T-beta: [0, 1) -> [0, 1) be the beta-transformation defined by T(beta)x = beta x mod1. We study the uniform recurrence properties of the orbit of a point under the beta-transformation to the point itself. The size of the set of points with prescribed uniform recurrence rate is obtained. More precisely, for any 0 <= r <= +infinity, the set @@@ {x is an element of[0,1): for all N >> 1, there exists 1 <= n <= N, s.t. vertical bar T(beta)(n)x-x vertical bar <= beta-((r) over capN)} @@@ is of Haufdorff dimension (1-(r) over cap /1+(r) over cap) if 0 <= (r) over cap <= 1 and is contable if (r) over cap >1. In addition, when (r) over cap = 1, it is uncountable but has zero Hausdorff dimension.

• 单位
  广东金融学院