Modified shrinking target problem for Matrix Transformations of Tori

Abstract: In this paper, 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 i m e s d

non-singular matrix with real coefficients. Then,

T

determines a self-map of the

d

-dimensional torus

m a t h b b T^{d} : = m a t h b b R^{d} / m a t h b b Z^{d}

. Let

p s i : m a t h b b R^{+} o m a t h b b R^{+}

be a real positive function, we study the Hausdorff dimension of the set �egin{equation*} �ig{mathtt{x}in mathbb{T}^d: T^n(mathtt{x})in L(f_n(mathtt{x}),psi(n)) ext{ for infinitely many } nin mathbb{N}�ig}, end{equation*} where

L (f_{n} (m a t h t t x), p s i (n))

are hyperrectangles in

m a t h b b T^{d}

and

f_{n}

is a sequence of Lipschitz functions defined on

m a t h b b T^{d}

with a uniform Lipschitz constant. Moreover, we also give the Hausdorff dimension of the set �egin{equation*} Big{mathtt{x}in mathbb{T}^d: prod_{1leq ileq d}|T_{�eta_i}^n(x_i)-x_i| < psi(n) ext{ for infinitely many } nin mathbb{N}Big}, end{equation*} where the

is the standard

-transformation with

and

| c d o t |

is the usual metric in

m a t h b b T

.

Mathematics Subject Classification ID

Metric theory of other algorithms and expansions; measure and Hausdorff dimension (11K55) Fractals (28A80) Dimension theory of smooth dynamical systems (37C45)

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