Abstract: Let {X(t),t ≥ 0} be Brownian motion on the Sierpinski gasket, E is a compact subset of[0,+∞). In this paper,we determine when X(E) has k-multiple points and when not.We also obtain the exact Hausdorff dimension of k-multiple times.

Key words: Brownian motion on the Sierpinski gasket, k-multiple points, k-multiple times