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 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
Formula...
A complete description is given only when the matrix is diagonalizable over Q. In other cases a result is obtained for sufficiently large t. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of t which is piecewise of the form (At+B)/(Ct+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.
Formula...
A complete description is given only when the matrix is diagonalizable over Q. In other cases a result is obtained for sufficiently large t. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of t which is piecewise of the form (At+B)/(Ct+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.
| Original language | English |
|---|---|
| Pages (from-to) | 381-398 |
| Number of pages | 18 |
| Journal | Journal of the London Mathematical Society |
| Volume | 60 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - Oct 1999 |