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Natural equilibrium states for multimodal maps. (English) Zbl 1211.37031
This paper studies the thermodynamic formalism for a class \({\mathcal F}\) of real multimodal maps with negative Schwarzian derivative. The main result shows for \(f \in{\mathcal F}\) and \(t\) in a maximal parameter interval that there exists a unique equilibrium measure \(\mu_t\) for the geometric potential \(- t \log |Df|\), generalizing results of H. Bruin and G. Keller [Ergodic Theory Dyn. Syst. 18, No. 4, 765–789 (1998; Zbl 0916.58020)], Y. Pesin and S. Senti [J. Mod. Dyn. 2, No. 3, 397–430 (2008; Zbl 1159.37007)], and H. Bruin and M. Todd [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 4, 559–600 (2009; Zbl 1192.37051)]. To obtain their main result, the authors use the theory of inducing schemes [see H. Bruin, Commun. Math. Phys. 168, No. 3, 571–580 (1995; Zbl 0827.58015) and H. Bruin and M. Todd, Commun. Math. Phys. 283, No. 3, 579–611 (2008; Zbl 1157.82022)]. The authors also prove for \(f \in{\mathcal F}\) that the pressure function is \(C^1\), strictly convex, and strictly decreasing over the maximal parameter interval. Additional results study how first order phase transitions relate to the existence of absolutely continuous invariant probability measures.

MSC:
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
37Axx Ergodic theory
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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