Abstract
Let T:J \to J be an expanding rational map of the Riemann sphere acting on its Julia set J and f:J\to \mathbb{R} denote a Hölder continuous function satisfying f(x) > \log|T^\prime(x)| for all x in J. Then for any point z_0 in J define the set D_{z_0}(f) of ‘well-approximable’ points to be the set of points in J which lie in the Euclidean ball
B\bigg(y,\exp\bigg(-\sum_{i=0}^{n-1} f(T^iy)\bigg)\bigg)
for infinitely many pairs (y,n) satisfying T^n(y)=z_0. In our 1997 paper, we calculated the Hausdorff dimension of D_{z_0} (f). In the present paper, we shall show that the Hausdorff measure \mathcal{H}^s of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.
B\bigg(y,\exp\bigg(-\sum_{i=0}^{n-1} f(T^iy)\bigg)\bigg)
for infinitely many pairs (y,n) satisfying T^n(y)=z_0. In our 1997 paper, we calculated the Hausdorff dimension of D_{z_0} (f). In the present paper, we shall show that the Hausdorff measure \mathcal{H}^s of this set is either zero or infinite. This is in line with the general philosophy that all ‘naturally’ occurring sets of well-approximable points should have zero or infinite Hausdorff measure.
| Original language | English |
|---|---|
| Pages (from-to) | 1773-1782 |
| Number of pages | 10 |
| Journal | Ergodic Theory and Dynamical Systems |
| Volume | 22 |
| Issue number | 6 |
| DOIs | |
| Publication status | Published - 2002 |