The chameleon groups of Richard J. Thompson: automorphisms and dynamics 
Matthew G. Brin 

Publications Math\'ematiques de l'Institut des Hautes \'Etudes Scientifiques 
84 (1997), 5-33. 

Abstract:
The automorphism groups of several of Thompson's countable groups of 
piecewise linear homeomorphisms of the line and circle are computed
and it is shown that the outer automorphism groups of these groups are 
relatively small. These results can be interpreted as stability results for
certain structures of PL functions on the circle. Machinery is developed to 
relate the structures on the circle to corresponding structures on the line.