2023/04/15 by Yuan, Na, Wang, ShuaiLing
#11K55 #28A80 #37C45 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.07532
We calculate the Hausdorff dimension of the fractal set \\mathttx∈ \mathbbTd: ∏1≤ i≤ d|Tβin(xi)-xi| lt; ψ(n) for infinitely many n∈ ℕ\, where the Tβi is the standard βi-transformation with βi>1, ψ is a positive function on ℕ and |⋅| is the usual metric on the torus \mathbbT. 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 \mathbbTd:=ℝd / 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 \\mathttx∈ \mathbbTd: Tn(x)∈ L(fn(\mathttx), Ψ(n)) for infinitely many n∈ ℕ\, where L(fn(\mathttx, Ψ(n))) is a hyperrectangle and \fn\n≥ 1 is a sequence of Lipschitz vector-valued functions on \mathbbTd with a uniform Lipschitz constant.