Dynamical Diophantine approximation of beta expansions of formal Laurent series

Let Fq be a finite field with q elements and Fq((X−1)) be the field of the formal Laurent series with an indeterminant X. Let Tβ be the β-transformation defined on Fq((X−1)). This paper is concerned with the size of the following dynamically defined limsup set{x∈Fq((X−1)):‖Tβnx−y0‖