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

@article{Bruin2008EquilibriumSF,
  title={Equilibrium States for Interval Maps: Potentials with sup $\phi$ − inf $\phi$ < htop(f)},
  author={Henk Bruin and Mike Todd},
  journal={Communications in Mathematical Physics},
  year={2008},
  volume={283},
  pages={579-611}
}
  • H. Bruin, M. Todd
  • Published 7 August 2008
  • Mathematics
  • Communications in Mathematical Physics
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… 
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TLDR
This work generalizes the result of F. Hofbauer on piecewise monotonic maps of the interval to arbitrary piecewise invertible dynamical systems and gets a sufficient condition for these maps to have a finite number of invariant and ergodic probability measures with maximal entropy.
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