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Complex maps without invariant densities. (English) Zbl 1122.37037

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.

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
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