Existência de medidas invariantes para aplicações no intervalo com presença de pontos críticos e singularidades

Abstract

We prove the existence of invariant probability measures absolutely continuous with respect to Lebesgue measure. Here we work with a class of maps that we denote by F, this class consists of interval maps f : M ! M, having critical points and singularities more other properties. I must mention that one of the properties is the condition of summability along the critical orbit which will help to have important results for our work. The main result says, for each f 2 F there is a probability measure invariant absolutely continuous. To achieve this result, we prove an auxiliary theorem that is the existence of a countable partition I of open intervals of M, an map that called induced time : M ! N that is constant on the elements of the partition I, such that the map ˆ f : M ! M defined by ˆ f = f we call induced map, satisfies three important properties that are, expanding, summable variation and summable induced time. So throughout the work we focus on evidence these three properties. The important point is that the first two properties together with theorem A ensures the existence of a measure absolutely continuous probability ˆ f, finally using the third property together with proposition A, we get the existence of an absolutely continuous probability measure for our f.