Equilibrium States for Interval Maps: Potentials with sup φ − inf φ < _toph(f)

We study an inducing scheme approach for smooth interval maps to prove existence and uniqueness of equilibrium states for potentials φ with the ‘bounded range’ condition sup φ − inf φ < htop(f), first used by Hofbauer and Keller [HK]. We compare our results to Hofbauer and Keller’s use of Perron-Frobenius operators. We demonstrate that this ‘bounded range’ condition on the potential is important even if the potential is Hölder continuous. We also prove analyticity of the pressure in this context.