Blue compact dwarf galaxies (BCDs) form stars at, for their sizes, extraordinarily high rates. In this paper, we study what triggers this starburst and what is the fate of the galaxy once its gas fuel is exhausted. We select four BCDs with smooth outer regions, indicating them as possible progenitors of dwarf elliptical galaxies. We have obtained photometric and spectroscopic data with the FORS and ISAAC instruments on the VLT. We analyse their infra-red spectra using a full spectrum fitting technique which yields the kinematics of their stars and ionized gas together with their stellar population characteristics. We find that the_stellar_ velocity to velocity dispersion ratio (v/sigma) of our BCDs is of the order of 1.5, similar to that of dwarf elliptical galaxies. Thus, those objects do not require significant (if any) loss of angular momentum to fade into early type dwarfs. This finding is in discordance with previous studies, which however compared the stellar kinematics of dwarf elliptical galaxies with the gaseous kinematics of star forming dwarfs. The stellar velocity fields of our objects are very disturbed and the star-formation regions are often kinematically decoupled from the rest of the galaxy. These regions can be more or less metal rich with respect to the galactic body, and sometimes they are long lived. These characteristics prevent us from pinpointing a unique trigger of the star formation, even within the same galaxy. Gas impacts, mergers, and in-spiraling gas clumps are all possible star-formation ignitors for our targets.
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