Skip to main content Accessibility help
×
×
Home

THE SHRINKING TARGET PROBLEM FOR MATRIX TRANSFORMATIONS OF TORI

  • RICHARD HILL (a1) and SANJU L. VELANI (a2)
    • Published online by Cambridge University Press: 01 October 1999

Abstract

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝd/ℤd. 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 xX 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

formula here

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.

Copyright

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the London Mathematical Society
  • ISSN: 0024-6107
  • EISSN: 1469-7750
  • URL: /core/journals/journal-of-the-london-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed