Periodic boundary points for transcendental Fatou components

Abstract

[eng] This thesis is framed in the field of Complex Dynamics, which studies discrete dynamical systems generated by the iteration of holomorphic functions. More precisely, given a transcendent, integer or meromorphic function, we consider the discrete dynamical system generated by it. Then the complex plane is divided into two totally invariant sets: the Fatou set, where the dynamics are stable; and Julia's ensemble, its complement, where the dynamics are chaotic. The Fatou set is open and generally has infinite related components, called Fatou components, and they are periodic, pre-periodic, or wandering. One of the basic results in Complex Dynamics (demonstrated by Fatou and Julia for rational functions) is that the Julia set is the closure of the repulsing periodic points of the function. This result was generalized by Baker by integer functions, and by Baker, Kotus, and Lü by transcendent meromorphic functions. We note that, given an invariant Fatou component, then its boundary is an invariant closed subset of the Julia set. So, the next question arises naturally: given a meromorphic function, and it is a periodic Fatou component, are the periodic points dense at their boundary? Note that, although the periodic points are dense in the Julia set, a priori they could accumulate at the boundary from its complement, without being at the boundary For example, if the Fatou component is a rotation domain with a locally connected boundary, then there is no periodic point. However, F. Przytycki and A. Zdunik showed that, by rational functions, rotation domains (i.e. Siegel's disks and Herman's rings) are the only exceptions for which the periodic points are not dense at the boundary. In particular, they gave a positive answer to the previous question for attraction or parabolic basins of rational functions. The work of F. Przytycki and A. Zdunik already shows us that the answer to such an elementary question is far from simple. Indeed, an exhaustive study of the boundaries of such Fatou components (which may not be locally connected) is necessary, combining tools of dynamics, measurement theory and conformal analysis. In the particular case of simply connected attraction basins, the proof is based on the properties of the function at the boundary from the point of view of the theory of measurement and Lyapunov's exponents, as well as precise estimates of the distortion of the Riemann application and the finite Blaschke products in the unit circle, and Pesin's theory conforms. For components of Fatou not limited to transcendent functions, the situation is even more delicate, due to the presence of the essential singularity, and most of the above techniques cannot be applied. Moreover, since the boundary of the Fatou component is not compact, it is not compact, nor is the existence of periodic points on the boundary evident. In view of the above questions, and the existing previous work to understand the boundaries of transcendent Fatou components, the following conjecture naturally arises, which is a large open problem in transcendent dynamics. Let it be a meromorphic function, and let it be a simply connected periodic Fatou component, such that it is not univalent. Then, there is a periodic point on the border of such a component of Fatou. In addition, if it is an attractor or parabolic basin, or a doubly parabolic Baker domain, then the periodic points are dense at the boundary. This thesis should be understood as significant progress in proving the above conjecture. Indeed, we demonstrate the existence and density of periodic points at the boundary of Fatou components under very weak hypotheses in the postsingular set, together with additional results in relation to boundary dynamics, escape points and accessibility. During the thesis, new techniques have been demonstrated, such as estimates in the distortion of internal functions and Pesin's theory for transcendent functions.