Renormalisation-induced phase transitions for unimodal maps

The thermodynamical formalism is studied for renormalisable maps of the interval and the natural potential $-t\log \mathrm{\mid }Df\mathrm{\mid }$. Multiple and indeed infinitely many phase transitions at positive $t$ can occur for some quadratic maps. All unimodal quadratic maps with positive topological entropy exhibit a phase transition in the negative spectrum.