Ergodic theory of equilibrium states for rational maps.

*(English)*Zbl 0718.58035Let \(X\) be a compact metric space, \(T\) be a continuous map on \(X\) and f be a real continuous function defined over \(X\). The “pressure” \(P(T,f)\) can be defined as:
\[
P(T,f)=\sup (h_{\mu}(T)+\int_{X}f d\mu)
\]
where the supremum is taken over the set of \(T\) invariant probability measures \(\mu\) on \(X\) and \(h_{\mu}\) is the metric entropy of \(\mu\).

Fundamental problems are the existence and uniqueness of measures for which the supremum is attained, called equilibrium states. The authors give the answer in the following situation: \(X\) is the Riemann sphere, \(T\) is an analytic endomorphism (rational map) of degree \(\geq 2\), and \(f\) is a Hölder continuous function such that \(P(T,f)>\sup_{z\in X}f(z)\). In this case there exists a unique equilibrium state.

Actually this equilibrium state is equivalent (with a continuous density) to the unique conformal probability measure \(\nu\) with respect to the function \(\exp (P(T,f)-f).\) The paper also contains a detailed analysis of the spectral properties of the Perron-Frobenius operator associated with \(\nu\), namely: \[ P(\phi)(x)=\sum_{y=T(x)}\phi(x)\exp(f(y)-P(T,f)). \] It is shown that this operator is almost periodic on the set of continuous functions on the Julia set of T from which it follows that the dynamical system \((X,T,\mu)\) is exact.

Fundamental problems are the existence and uniqueness of measures for which the supremum is attained, called equilibrium states. The authors give the answer in the following situation: \(X\) is the Riemann sphere, \(T\) is an analytic endomorphism (rational map) of degree \(\geq 2\), and \(f\) is a Hölder continuous function such that \(P(T,f)>\sup_{z\in X}f(z)\). In this case there exists a unique equilibrium state.

Actually this equilibrium state is equivalent (with a continuous density) to the unique conformal probability measure \(\nu\) with respect to the function \(\exp (P(T,f)-f).\) The paper also contains a detailed analysis of the spectral properties of the Perron-Frobenius operator associated with \(\nu\), namely: \[ P(\phi)(x)=\sum_{y=T(x)}\phi(x)\exp(f(y)-P(T,f)). \] It is shown that this operator is almost periodic on the set of continuous functions on the Julia set of T from which it follows that the dynamical system \((X,T,\mu)\) is exact.

Reviewer: J.Lacroix (Villetaneuse)