THE SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

Abstract

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝ^d/ℤ^d. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x ∈ X such that Tⁿ(x) ∈ B(n) for infinitely many n ∈ ℕ. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity

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A complete description is given only when the matrix is diagonalizable over ℚ. In other cases a result is obtained for sufficiently large τ. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of τ which is piecewise of the form (Aτ+B)/(Cτ+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.