Bruin, Henk; Todd, Mike Complex maps without invariant densities. (English) Zbl 1122.37037 Nonlinearity 19, No. 12, 2929-2945 (2006). In this interesting paper, the authors consider complex polynomials \(f(z)= z^1+ c\) for \(1\in 2\mathbb{N}\) and \(c\in\mathbb{R}\) and find some combinatorial types and values of 1 such that there is no invariant probability measure equivalent to the conformal measure on the Julia set. The Feigenbaum and Fibonacci maps are considered, which are interesting examples in the measure theoretic context, especially when the critical order 1 is large. Reviewer: Alois Klíč (Praha) Cited in 5 Documents MSC: 37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems 37F25 Renormalization of holomorphic dynamical systems Keywords:invariant probability measure; Hausdorff dimension; dynamical dimension; Feigenbaum maps; Fibonacci maps; Koebe space PDFBibTeX XMLCite \textit{H. Bruin} and \textit{M. Todd}, Nonlinearity 19, No. 12, 2929--2945 (2006; Zbl 1122.37037) Full Text: DOI arXiv Link