Hyperbolic dimension and Poincaré critical exponent of rational maps (Q1702459)

The authors study the relationship between the Poincaré critical exponent of rational maps and the hyperbolic dimension. They define the set \(\mathrm{Crit}'(f) = \mathrm{Crit}(f) \cap J(f)\), where \(f\) is a rational map of degree at least two of the Riemann sphere, \(J(f)\) is the Julia set of \(f\) and \(\mathrm{Crit}(f)\) is the set of critical points of \(f\). The post-critical set \(P(f)\) is the closure of the set \(PC(f)\), which is the union of the forward orbits of all critical points in \(J(f)\). Investigating the properties of the canonical Julia set and the dissipative measure, the authors prove their main result: let \(f: \overline{\mathbb{C}} \to \overline{\mathbb{C}} \) be a rational map of degree at least two. Assume that for every critical point \(c \in \mathrm{Crit}'(f)\) we have \(\lim \sup_{n \to \infty}|Df^n(f(c))| > 0\), and suppose that \(J(f) \setminus P(f) \neq 0\). Then the Poincaré critical exponent of \(f\) is equal to the hyperbolic dimension of \(f\).