Invariant measures for critical covering maps of the circle

Author(s)

Tiago Estrela de Oliveira

• Document type: Doctoral Thesis
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI)
• Defense date: 2014-05-05
• Advisor: Edson Vargas

Abstract

The main goal of this work is to describe some measure-theoretic aspects of the dynamical systems generated by covering maps of the circle of the circle which has a critical point, called critical covering map of the circle. In the most interesting situation, this covering map, say f, has no attracting periodic orbit and, it is topologically conjugate with a covering map of circle without critical point and preserving the Lebesgue measure. Then, it is natural to imagine that f has an invariant probability measure which is absolutely continuous with respect the Lebesgue measure (for short, acip). Nevertheless, we prove the existence of an analytic critcal covering map of the circle, without attracting periodic orbits and without acip. To get this result we construct a critical covering map whose critical point is strongly recurrent. The power of recurrence of the a critical point is measured combinatorially and this is a very delicate problem in the present context. One of the main difficulty comes from the lack of a natural and dynamicaly defined simmetry around the critical point. Due to this, the tools used to treat similar problems in the context of unimodal maps have to be adapted or even changed in drastic way. As a final result, besides the existence of analy- tic critical covering maps of the circle without attracting periodic orbits and without acip Como resultado final, we indicate the types of combinatorics which lead to such behavior. As a byproduct of all of this, one have a combinatorial way to measure how strong is the recurrence of the critical point. (AU)

• FAPESP's process: 09/08367-5 - Physical measures for critical covering maps of the circle
• Grantee: Tiago Estrela de Oliveira
• Support Opportunities: Scholarships in Brazil - Doctorate