arXiv subjects

Helena Mihaljevic-Brandt

Publications and source records attributed to Helena Mihaljevic-Brandt.

3 recordsLinked to original sources

arXiv2009

Dynamical approximation and kernels of nonescaping-hyperbolic components

Let F_n be families of entire functions, holomorphically parametrized by a complex manifold M. We consider those parameters in M that correspond to nonescaping-hyperbolic functions, i.e., those maps f in F_n for which the postsingular set P(f) is a compact subset of the Fatou set F(f) of f. We prove that if F_n converge to a family F in the sense of a certain dynamically sensible metric, then every nonescaping-hyperbolic component in the parameter space of F is a kernel of a sequence of nonescaping-hyperbolic components in the parameter spaces of F_n. Parameters belonging to such a kernel do not always correspond to hyperbolic functions in F. Nevertheless, we show that these functions must be J-stable. Using quasiconformal equivalences, we are able to construct many examples of families to which our results can be applied.

math.DS↗

arXiv2009

Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that J(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.

math.DS↗

arXiv2009

A landing theorem for dynamic rays of geometrically finite entire functions

A transcendental entire function f is called geometrically finite if the intersection of the set of singular values with the Fatou set is compact and the intersection of the postsingular set with the Julia set is finite. (In particular, this includes all entire functions with finite postsingular set.) If f is geometrically finite, then the Fatou set of f is either empty or consists of the basins of attraction of finitely many attracting or parabolic cycles. Let z_0 be a repelling or parabolic periodic point of such a map f. We show that, if f has finite order, then there exists an injective curve consisting of escaping points of f that connects z_0 to infinity. (This curve is called a dynamic ray.) In fact, the assumption of finite order can be weakened considerably; for example, it is sufficient to assume that f can be written as a finite composition of finite-order functions.

math.DS↗