  We show that novel fixed points (characterized by matrices which specify the
splitting of the currents at the junction) can be accessed in a system which
contains a junction of three quantum Hall line junctions. For such a junction
of fractional quantum Hall edge states, we find that it is possible for both
the flower (single droplet) and islands (three droplets) configurations to be
stable in an intermediate region, for a range of values of the inter-edge
repulsive interactions. A measurement of the tunneling conductance as a
function of the gate voltage controlling inter-edge repulsions can give a clear
experimental signal of this region.
