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Ergodic theory of equilibrium states for rational maps. (English) Zbl 0718.58035
Let $$X$$ 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.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 28D20 Entropy and other invariants 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
##### Keywords:
equilibrium states; Julia set
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