CENTRAL LIMIT THEOREM FOR THE MODULUS OF CONTINUITY OF AVERAGES OF OBSERVABLES ON TRANSVERSAL FAMILIES OF PIECEWISE EXPANDING UNIMODAL MAPS

Author(s)

De Lima, Amanda ^[1] ; Smania, Daniel ^[1]

Affiliation

^[1] Univ Sao Paulo, ICMC, Dept Matemat, Caixa Postal 668, BR-13560970 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU; v. 17, n. 3, p. 673-733, JUN 2018.
• Web of Science Citations: 1

Abstract

Consider a C-2 family of mixing C-4 piecewise expanding unimodal maps t is an element of {[}a, b] bar right arrow f(t), with a critical point c, that is transversal to the topological classes of such maps. Given a Lipchitz observable consider the function R-phi(t) = integral phi d mu(t), where mu(t) is the unique absolutely continuous invariant probability of f(t). Suppose that sigma(t) > 0 for every t is an element of {[}a, b], where sigma(2)(t) = sigma(2)(t)(phi) = lim(n ->infinity) integral (Sigma(n-1)(j=0) (phi o f(t)(j) - integral phi d mu(t))/root n)(2) d mu(t). We show that m [ t is an element of{[}a, b]: t + h is an element of {[}a, b] and 1/psi(t)root-log vertical bar h vertical bar (R-phi(t + h) - R-phi(t)/h) <= y ] converges to 1/root 2 pi integral(y)(-infinity) e(-s2/2) ds, where psi(t) is a dynamically defined function and m is the Lebesgue measure on {[}a, b], normalized in such way that m({[}a, b]) = 1. As a consequence, we show that R-phi is not a Lipchitz function on any subset of {[}a, b] with positive Lebesgue measure. (AU)

• FAPESP's process: 10/17419-6 - Transversal families of piecewise expanding maps
• Grantee: Amanda de Lima
• Support Opportunities: Scholarships in Brazil - Doctorate