Measure rigidity for some transcendental meromorphic functions
The author considers meromorphic functions \(f\) of the form \(f(z)=R(e^z)\) where \(R\) is a rational function such that \(\infty\) is an asymptotic value of \(f\). Let \(f\) and \(g\) be hyperbolic meromorphic functions of this form which are topologically conjugate on their Julia sets; that is, there exists a homeomorphism \(h:J(f)\to J(g)\) such that \(h\circ f=g\circ h\) on \(J(f)\backslash f^{-1}(\infty)\). Here \(J(\cdot)\) denotes the Julia set.NEWLINENEWLINEAssuming that \(f\) and \(g\) are non-exceptional in a certain sense, conditions are given that imply that \(h\) extends to an affine map conjugating \(f\) to \(g\). For example, it is shown that this is the case if \(h\) and \(h^{-1}\) are Lipschitz continuous or if \(h\) preserves the moduli of multipliers of periodic points; that is, if \(f^p(z)=z\), then \(|(f^p)'(z)|=|(g^p)'(h(z))|\).
scientific article; zbMATH DE number 6052999
scientific article