Equilibrium states of interval maps for hyperbolic potentials (Q2877442)

Let \(I\) be a compact interval in \({\mathbb{R}}\) and \({f:I\to I}\) be a differentiable map with additional properties: 1) \(f\) is multimodal; 2) \(Df\) is Hölder continuous; 3) the set of critical points of \(f\) is finite; and 4) the Julia set~\(J(f)\) contains at least 2 points and is completely invariant. The authors study the thermodynamic formalism of such map \(f\) for a Hölder continuous potential \(\varphi\) which is hyperbolic for \(f\), i.e., satisfies the condition NEWLINE\[NEWLINE \sup_I\frac{1}{n}\sum_{j=0}^{n-1}\varphi\circ f^j<P(f,\varphi), NEWLINE\]NEWLINE where \(P(f,\varphi)\) is the pressure. In particular, the authors prove the existence of a Borel probability atom-free \({\exp{(P(f,\varphi)-\varphi)}}\)-conformal measure \(\mu\) for \(f\) on \(J(f)\) and a unique equilibrium state \(\nu\) of \(f\) absolutely continuous with respect to \(\mu\) for the potential \(\varphi.\)