Lyapunov spectrum for exceptional rational maps (Q2860888)

The authors study in this paper the complexity of the Lyapunov exponent function of rational maps on \(\mathbb{C}\cup\{\infty\}\). They show that the dimension spectra of the Lyapunov regular and irregular sets can be characterized in terms of the Legendre transformation of the hidden pressure (defined by the variational principle over all non-atomic invariant measures).NEWLINENEWLINEThis is an extension of their previous work [the authors, Math. Ann. 348, No. 4, 965--1004 (2010; Zbl 1206.37016)], where similar results have been obtained for the following three cases: {\parindent=6mm \begin{itemize}\item[(1)] \(\text{Crit}(f)\cap J(f)=\emptyset\), \item[(2)] \(f\) is non-exceptional, \item[(3)] \(f\) is exceptional, but \(\Sigma(f)\cap J(f)=\emptyset\). NEWLINENEWLINE\end{itemize}} Here \(J(f)\) is the Julia set of \(f\), \(\text{Crit}(f)\) is the set of critical points of \(f\), and \(\Sigma(f)\) is the set used to define the exceptional property of \(f\). That is, a rational function \(f\) is said be \textit{exceptional}, if there exists a nonempty, finite and forward invariant subset \(\Sigma'\subset J(f)\), such that \(f^{-1}\Sigma'\backslash \Sigma'\subset \text{Crit}(f)\). Then \(\Sigma(f)\) is a maximal subset with this property.