Modified Shrinking Target Problem For Matrix Transformations Of Tori

In this paper, we calculate the Hausdorff dimension of the fractal set x ∈ 𠕋d :∠1≤i≤d|Tβin(x i) − xi| < ψ(n) for infinitely many n ∈ ℕ , where Tβi is the standard βi-transformation with βi > 1, ψ is a positive function on ℕ and |⋅| is the usual metric on the torus 𠕋. Moreover, we investigate a modified version of the shrinking target problem, which unifies the shrinking target problems and quantitative recurrence properties for matrix transformations of tori. Let T be a d × d non-singular matrix with real coefficients. Then, T determines a self-map of the d-dimensional torus 𠕋d := ℠d/ℤd. For any 1 ≤ i ≤ d, let ψi be a positive function on ℕ and Ψ(n) := (ψ1(n),…,ψd(n)) with n ∈ ℕ. We obtain the Hausdorff dimension of the fractal set {x ∈ 𠕋d : Tn(x) ∈ L(f n(x), Ψ(n)) for infinitely many n ∈ ℕ}, where L(fn(x, Ψ(n))) is a hyperrectangle and {fn}n≥1 is a sequence of Lipschitz vector-valued functions on 𠕋d with a uniform Lipschitz constant.