Fine inducing and equilibrium measures for rational functions of the Riemann sphere
Let \(f:\widehat{\mathbb C}\rightarrow\widehat{\mathbb C}\) be a rational map of degree \(\deg(f) \geq 2\). The Julia set of \(f\) is denoted by \(J(f)\). Let \(C(J(f))\) be a Banach space of real-valued continuous functions on it equipped with supremum norm. The elements of \(C(J(f))\) are called potentials. If \(n\in\mathbb N^\ast\) and \(\varepsilon>0\) we say a finite subset \(E\) of \(J(f)\) is \((n,\varepsilon)\)-separated if whenever \(z \neq z' \in E\), there exists some \(i\in\{0,\cdots,n-1\}\) and \(|f^i(z)-f^i(z')|>\varepsilon\), where \(f^i=\underbrace{f\circ f \circ \cdots \circ f}_{i\mathrm{ times}}\). For \(\phi \in C(J(f))\) define \[ P_n(\phi,\varepsilon)=\sup\left\{\sum_{z \in E}\exp\left(\sum_{i=1}^{n-1}\phi(f^i(z))\right)/ E \text{ is }(n,\varepsilon)\mathrm{-separated}\right\}, \] and put \(P(\phi,\varepsilon)=\limsup_{n \rightarrow +\infty}P_n(\phi,\varepsilon)\). The topological pressure of \(\phi\) is given by \[ P(\phi)=\lim_{ \varepsilon \rightarrow 0}P(f,\phi,\varepsilon). \] By the celebrated Variational Principle we have \[ P(\phi) =\sup_{\{\mu:f_{\star}\mu=\mu\}}\left\{h_{\mu}(f)+\int \phi d\mu\right\}, \] where \(h_{\mu}(f)\) is the Kolmogorov-Sinai entropy and \(f_{\star}\mu\) is the pushforward measure of \(\mu\) under \(f\). The measures for which \(h_{\mu}(f)+\int \phi d\mu=P(\phi)\) are called equilibrium states. Let \(\phi\) be a Hölder-continuous function such that \(P(\phi)>\sup(\phi)\). It is known that there exists a unique equilibrium state measure \(\mu_{\phi}\) for this potential. In the paper under review, the authors characterize all maps and potentials for which \(\mathrm{HD}(\mu_{\phi})=\mathrm{HD}(J(f))\), where \(\mathrm{HD}\) is the Hausdorff dimension. Consequently, they obtain that \(\mathrm{HD}(\mu_{\phi})=2\) if and only if the potential is cohomologous to a constant function in \(C(J(f))\) and \(f\) is a critically finite rational map with a parabolic orbifold. They further establish that the map \(t \mapsto P(t\phi)\) is real-analytic and they investigate finer geometric and stochastic properties, namely, exponential decay of correlations, the central limit theorem, and the law of iterated logarithm for Hölder potentials.
scientific article; zbMATH DE number 6509701
scientific article