MODIFIED SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

Na Yuan () and Shuailing Wang
Additional contact information
Na Yuan: School of Mathematics, Guangdong University of Education, Guangzhou 510303, P. R. China
Shuailing Wang: Department of Mathematics, South China University of Technology, Guangzhou 510641, P. R. China

FRACTALS (fractals), 2024, vol. 32, issue 05, 1-10

Abstract: 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.

Keywords: Shrinking Target Problem; Fractal Sets; Hausdorff Dimension (search for similar items in EconPapers)
Date: 2024
References: Add references at CitEc
Citations:

Downloads: (external link)
http://www.worldscientific.com/doi/abs/10.1142/S0218348X24500762
Access to full text is restricted to subscribers

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:wsi:fracta:v:32:y:2024:i:05:n:s0218348x24500762

Ordering information: This journal article can be ordered from

DOI: 10.1142/S0218348X24500762

Access Statistics for this article

FRACTALS (fractals) is currently edited by Tara Taylor

More articles in FRACTALS (fractals) from World Scientific Publishing Co. Pte. Ltd.
Bibliographic data for series maintained by Tai Tone Lim ().