arXiv subjects

Monica Moreno Rocha

Publications and source records attributed to Monica Moreno Rocha.

2 recordsLinked to original sources

arXiv2026

Puzzle Pieces, Bi-accessibility, and Connectivity of the Julia Set for Generalized Blaschke Products

We study the dynamics of a parametric family of rational functions of odd degree, where each function is a generalized Blaschke product that maps the unit circle onto itself. The action of the Blaschke product restricted to the unit circle defines a circle map, and the parameter space of the family exhibits Arnold tongues. As the parameter varies over an Arnold tongue, the action of the circle map changes from a diffeomorphism to a non-injective endomorphisms. Using a combinatorial study of puzzle pieces, we show that for adjacent parameters inside the Arnold tongues, there exist bi-accessible repelling cycles. This topological feature enables us to exclude the presence of multiply connected Fatou components whenever Herman rings are absent. As a result, we obtain a complete characterization of the connectivity of the Julia set for each member of the parametric family.

math.DS↗

arXiv2011

Non-landing hairs in Sierpinski curve Julia sets of transcendental entire maps

We consider the family of transcendental entire maps given by $f_a(z)=a(z-(1-a))\exp(z+a)$ where $a$ is a complex parameter. Every map has a superattracting fixed point at $z=-a$ and an asymptotic value at $z=0$. For $a>1$ the Julia set of $f_a$ is known to be homeomorphic to the Sierpiński universal curve, thus containing embedded copies of any one-dimensional plane continuum. In this paper we study subcontinua of the Julia set that can be defined in a combinatorial manner. In particular, we show the existence of non-landing hairs with prescribed combinatorics embedded in the Julia set for all parameters $a\geq 3$. We also study the relation between non-landing hairs and the immediate basin of attraction of $z=-a$. Even as each non-landing hair accumulates onto the boundary of the immediate basin at a single point, its closure, nonetheless, becomes an indecomposable subcontinuum of the Julia set.

math.DS↗