The finer geometry and dynamics of the hyperbolic exponential family (Q1411306)

The paper contains a study of the maps \(f_\lambda(z) = \lambda e^z\), where \(\lambda\) is a nonzero complex number. It concentrates on the subset \(J_r(f_\lambda)\) of the Julia set of \(f_\lambda\) consisting of points whose forward iterates by \(f_\lambda\) do not go to infinity. A detailed proof is given that, if \(0<\lambda < 1/e\), then the Hausdorff dimension of \(J_r(f_\lambda)\) is less than 2. The importance of the restriction on \(\lambda\) is that it ensures that \(f_\lambda\) has an attracting fixed-point. The changes necessary to extend the result to \(f_\lambda\) with an attracting periodic orbit are outlined. The argument goes by using \(f_\lambda\) to induce a map \(F\) from some horizontal strip of height \(2\pi i\) to itself (using the fact that \(f_\lambda\) has period \(2\pi i\)). It is shown that there is a unique conformal probability measure for \(F\) and a \(\sigma\)-finite conformal measure for \(F_\lambda\). A recent result of M. Martens is used to show the existence of a unique ergodic Borel probability measure for \(F\) that is equivalent with the conformal measure.