Gibbs state for some class of meromorphic functions (Q2385792)

Let \(\mathcal{F}\) denote the class of transcendental meromorphic functions \(F\) in the complex plane of the form \(F(z) = H(e^z)\), where \(H\) is a nonconstant rational function. The set of singularities of \(F\) consists of finitely many critical values and two asymptotic values \(H(0)\), \(H(\infty)\). The subclass \(\mathcal{F}_1\) consists of those functions \(F \in \mathcal{F}\) with \(H(0) \neq \infty\), \(H(\infty) \neq \infty\) and \(\text{dist}_\chi{(P_1(F),J_F)}>0\), where \(\chi\) is a chordal metric, \(J_F\) is the Julia set of \(F\) and \(P_1(F) := \overline{\bigcup_{n=0}^\infty F^n(\{H(0),H(\infty)\})}\). Since \(F\) is \(2\pi i\)-periodic, it is usual to consider \(F\) on the cylinder which is the quotient space \(\mathcal{P} = \mathbb{C}/\sim\). If \(\pi : \mathbb{C} \to \mathcal{P}\) denotes the canonical projection, then \(F\) projects down to a holomorphic map \(f : \mathcal{P} \setminus \pi(F^{-1}(\infty)) \to \mathcal{P}\) so that \(f \circ \pi = \pi \circ F\). The Julia set of \(f\) is then defined by \(J_f := \pi(J_F \cap \mathbb{C})\). On \(\mathcal{P}\), there is introduced a class of summable potentials \(\varphi\) and a transfer operator \(\mathcal{L}_\varphi\) which acts on the Banach space \(C(J_f)\) of continuous functions on \(J_f\). For \(z \in \mathcal{P}\) set \[ P_z(\varphi) := \limsup_{n\to\infty}{\frac{1}{n}\,\log{\mathcal{L}_\varphi^n 1(z)}} = \limsup_{n\to\infty}{\frac{1}{n}\,\log{\sum_{y \in f^{-1}(z)} \exp{\left(\varphi(y)+\dotsb+\varphi(f^{n-1}(y))\right)}}}\,. \] Now, let \(F \in \mathcal{F}_1\), \(f:=\pi(F)\), and assume that \(\varphi\) is a summable potential such that \(\sup{\varphi} < \sup_{z \in J_f}{P_z(\varphi)}\). Then the author proves the following assertions. (1) The limit \(P(\varphi) = \lim_{n\to\infty}{\frac{1}{n}\,\log{\mathcal{L}_\varphi^n 1(z)}}\) exists and is independent of \(z \in J_f\). (2) There exists a unique \(\exp{(P(\varphi)-\varphi)}\)-conformal measure \(m_\varphi\) on \(J_f\) and a unique Borel probability \(f\)-invariant measure \(\mu_\varphi\) absolutely continuous with respect to \(m_\varphi\). The measure \(\mu_\varphi\) is equivalent to \(m_\varphi\), \(h = \frac{d\mu_\varphi}{dm_\varphi}\), the normalized fixed point of the Perron-Frobenius operator \(\Hat{\mathcal{L}}_\varphi\), and is called the Gibbs state of the summable potential \(\varphi\).