The recurrence set arising in α-Lüroth transformation

Let α={An}n≥1α={An}n≥1 be a sequence of left-open and right-closed intervals which partition (0,1](0,1]. The α  -Lüroth transformation LαLα is defined as an infinite piecewise linear map which maps AnAn linearly onto (0,1](0,1] for every n≥1n≥1. Then every point x∈(0,1]x∈(0,1] is attached with a finite or infinite integer sequence {ℓn}n≥1{ℓn}n≥1 by looking at the coding of its trajectory. In this note, we consider the size of the recurrent set in such a system. More precisely, let x0∈(0,1]x0∈(0,1] with an infinite α  -Lüroth expansion, and {tn}n≥1{tn}n≥1 an arbitrary non-decreasing sequence of natural numbers. The recurrence set of α  -Lüroth transformation LαLα is defined asF(x0)={x∈[0,1]:Lαn(x)∈A(ℓ1,ℓ2,⋯,ℓtn)(x0),F(x0)={infinitely oftenn∈N}, where A(ℓ1,ℓ2,⋯,ℓtn)(x0)A(ℓ1,ℓ2,⋯,ℓtn)(x0) denotes the tntn-th cylinder containing x0x0 in α  -Lüroth expansion. The Hausdorff dimension of F(x0)F(x0) is obtained.