Semihyperbolic Transcendental Semigroups

Singularities of Schröder Maps and Unhyperbolicity of Rational Functions

We study transcendental singularities of a Schröder map arising from a rational function f , using results from complex dynamics and Nevanlinna theory. These maps are transcendental meromorphic functions of finite order in the complex plane. We show that their transcendental singularities lie over the set where f is not semihyperbolic (unhyperbolic). In addition, if they are direct, then they l...

The Regularity of Semihyperbolic Patches Near Sonic Lines for The 2-D Euler System in Gas Dynamics

We study the regularity of semihyperbolic patches of self-similar solutions near sonic lines to a Riemann problem for the two-dimensional (2-D) Euler system. As a result, it is verified that there exists a global solution in the semihyperbolic patch up to the sonic boundary and that the sonic boundary has C1-regularity. The study of the semihyperbolic patches of solutions for the Euler system w...

Analytic Families of Semihyperbolic Generalized Polynomial-like Mappings

We show that the Hausdorff dimension of Julia sets in any analytic family of semihyperbolic generalized polynomial-like mappings (GPL) depends in a real-analytic manner on the parameter. For the proof we introduce abstract weakly regular analytic families of conformal graph directed Markov systems. We show that the Hausdorff dimension of limit sets in such families is real-analytic, and we asso...

Quasi-flats in Semihyperbolic Groups

We prove that the Cayley graph of a group which is semihyperbolic but not hyperbolic contains a subset quasi-isometric to R2.

Real Analyticity of Topological Pressure for Parabolically Semihyperbolic Generalized Polynomial-like Maps