# Dynamical Day

October 14th,2019

The aim of this meeting is to gather a group of mathematicians working on dynamical systems to discuss new developments in the area. It will be held at Sala C506, in the Construcción Civil building, 5th floor, Campus San Joaquín, UC.

Jairo Bochi

Italo Cipriano

Godofredo Iommi

Jan Kiwi

Mario Ponce

14:30-15:30 Victor Nopal Coello

15:45-16:45 Van Tu Le

16:50-17:50 Pablo Aguirre

TITLES & ABSTRACTS

Speaker: Victor Nopal

Title: Construction of Herman n-rings from Siegel disk on C_p

Abstract: Let R be a rational map of degree dgeq 2 with coefficients in the non-Archimedean field C_p, with p a prime number. Assume that the Fatou set F_R associated to R contains an m-cycle of Siegel disk, say D_1, D_2, …, D_m. In this talk I will explain how to build a racional map Q of degree d+1 with coefficients on C_p and such that F_Q contains an m-cycle of Herman n-ring coming from the Siegel disks of F_R.

Speaker: Van Tu Le

Title: Fixed points of post-critically algebraic endomorphisms

Abstract: A holomorphic endomorphism of $mathbb{CP}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map at its fixed points. When n=1, a well-known fact is that the eigenvalue at a fixed point is either superattracting or repelling. We prove that when n=2 the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson. When n≥2 and the fixed point is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one which was already obtained by Fornaess and Sibony under a hyperbolicity assumption on the complement of the post-critical set.

Speaker: Pablo Aguirre

Title: Nonlinear dynamics of propagation and containment of dengue

Abstract: Arboviruses such as dengue, zyka and chikungunya are viruses transmitted to humans by mosquitoes. In particular, Aedes Aegypti mosquito is the responsible for dengue transmission. In the absence of medical treatments and vaccines, one of the control methods is to introduce Aedes Aegypti mosquitoes infected by the bacterium Wolbachia into a population of wild (uninfected) mosquitoes. The goal consists in achieving population replacement in finite time by driving the population of wild mosquitoes towards extinction while keeping Wolbachia-infected mosquitoes alive. This strategy has several advantages for control of dengue: Wolbachia decreases the virulence of the dengue infection and it reduces the lifespan of the mosquito. Moreover, mating of a female uninfected by Wolbachia and an infected male leads to sterile eggs. We consider a competition model between wild Aedes Aegypti female mosquitoes and those infected with the bacteria Wolbachia in the form of a system of nonlinear differential equations. Our goal is to examine the basin of attraction of a desired equilibrium state. For this purpose, we study how the stable manifold that forms the basin boundary of interest changes under parameter variation. To achieve this, we combine analytical tools from dynamical systems and geometric singular perturbation theory with numerical continuation methods. This allows us to present a strategy to get the desired population replacement with a minimum amount of released infected mosquitoes in a human, external intervention by choosing an appropriate combination of initial conditions and parameter values.

This event is supported by CONICYT through the proyecto Anillo ACT172001 «New trends in ergodic theory» and by the Facultad de Matemáticas, Pontificia Universidad Católica de Chile