Constant Wattage Autotransformer (Ballast)

Can anyone tell me how the current is regulated for these types of transformers. I was thinking it had something to do with the magnetics of the circuit. As with any ballasts, it needs to control the current or else it will go up exponentially because the resistance of an arc decreases as it gets hotter. The transformers consists of two coils in a core that looks likes this:
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The coils are wound around the center and iron shunts are placed between them. There is a slit in the core where the secondary coil wraps around, however, the core is still connected. I believe they are trying to saturate core at that part, but I am not sure. Any suggestions?

 

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Ballasts are reactors with positive voltage-current characteristic compensating the arc negative performance. That could be done with resistors but resistors waste energy while reactors do not. Besides, reactors allow for arc lightening. Frequently, ballasts combines reactor and trasformer, for voltage adaptation, the reactance is then obtained by mean of a high leakeage flux between both windings.
Julian