Arguing from self similarity, Julian found In[240]:= Drag /@ {13/30, 21/30, 23/30} Out[240]= {1/2 + I/6, 1/2 + I/6, 1/2 + I/6} i.e., the function visits 1/2+i/6 three times, where Drag is the exact rational definition I sent recently. By dragon/2 ~ dragon, In[90]:= Drag /@ ({13/30, 21/30, 23/30}/2) Out[90]= {1/6 + I/3, 1/6 + I/3, 1/6 + I/3} and In[91]:= Drag /@ (1 - {13/30, 21/30, 23/30}/2) Out[91]= {2/3 + I/6, 2/3 + I/6, 2/3 + I/6} In[92]:= Drag /@ ((1 - {13/30, 21/30, 23/30}/2)/2) Out[92]= {1/4 + (5 I)/12, 1/4 + (5 I)/12, 1/4 + (5 I)/12} By iterating #/2 and 1-#/2, we can construct a dense set. After one iteration, all three inverse images will be on the same side of 1/2, but Julian points out there are other cases with denominator 30*2^n, e.g., In[99]:= Drag /@ {199/480, 71/160, 329/480} Out[99]= {11/24 + (7 I)/24, 11/24 + (7 I)/24, 11/24 + (7 I)/24} There appear to be no quadruple points. We found a slightly questionable proof that all spacefilling functions are dense with triple points. (Lemma: They map closed intervals onto closed sets.) By methodical search, Julian found another family of denominators: In[93]:= Drag /@ {13/28, 17/28, 19/28} Out[93]= {3/5 + (3 I)/10, 3/5 + (3 I)/10, 3/5 + (3 I)/10} In[94]:= Drag /@ ({13/28, 17/28, 19/28}/2) Out[94]= {3/20 + (9 I)/20, 3/20 + (9 I)/20, 3/20 + (9 I)/20} We did not find any families of denominators besides 28*2^n and 30*2^n. --rwg PS, while investigating the (three valued) inverse dragon function, Neil made the lovely monochrome image: http://gosper.org/draglarge.png