Geometry and dynamics of some meromorphic functions
نویسنده
چکیده
The meromorphic maps fλ(z) = λ(1−exp(−2z))−1, λ > 0, of the complex plane are thoroughly investigated. With each map fλ associated is its projection Fλ on the infinite cylinder Q. This map and the set Jr(Fλ) consisting of those points in the cylinder Q whose ω-limit set under Fλ is not contained in the set {0,−∞} will form the primary objects of our interest in this article. Let hλ = HD(Jr(Fλ)) be the Hausdorff dimension of Jr(Fλ). We prove that hλ ∈ (1, 2). The hλ-dimensional Hausdorff measure Hhλ of Jr(Fλ) is proven to be positive and finite. The hλ-dimensional packing measure of Jr(Fλ) is shown to be locally infinite at every point of this set. There exists a unique Borel probability Fλ-invariant measure μλ on Jr(Fλ) absolutely continuous with respect to the Hausdorff measure Hhλ . This measure turns out to be ergodic and equivalent to Hhλ .
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