Transversal families of piecewise expanding maps

Author(s)

Amanda de Lima

• Document type: Doctoral Thesis
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2015-05-07

Examining board members

Daniel Smania Brandão; Eduardo Garibaldi; Alexander Eduardo Arbieto Mendoza; Ali Tahzibi

• Advisor: Daniel Smania Brandão

Abstract

Let t:[a,b] → ft be a C2 family of \"good\" C4 e piecewise expanding unimodal maps, with a critical point c, that is transversal to the topological classes of such maps. Given a lipchitzian observable ∅, consider the function ℛ∅(t)=∫∅dµt, where µt is the unique bsolutely continuous invariant probability of ft. We show a central limit theorem for the modulus of continuity of ℝ∅, that is limh→0m{t ∈ [a,b] : t + h ∈ [a,b] e 1/(Ψ(t)(-log|h|)½)((ℛ∅(t + h) - ℛ∅(t))/h) ≤ y} converges to 1/(2π)½ ∫y-∞e-s2/2ds. Now, let us consider a C2+ε expanding map f : 𝕊1 → 𝕊1 and a C1+ε periodic function v : 𝕊1 → ℝ. We show that the unique bounded solution of the twisted cohomological equation v(x) = α(f(x)) - Df(x)α(x) is either of class C1+ε or nowhere differentiable. We also prove that if α is nowhere differentiable, them the modulus of continuity of α satisfies a central limit theorem, that is, there is α &gt 0 such that limh→0µ{x : (α(x + h) - α(x))/(σ𝓁h(-log|h|)½) ≤ y} = 1/(2π)½ ∫y-∞e-t2/2dt, where µ is the absolutely continuous invariant probability of f. (AU)

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