Aline Melo
Aline Melo
Postdoctoral Researcher at the Federal University of Ceará
Postdoctoral Researcher at the Federal University of Ceará
My main interests in Mathematics are focused on Dynamical Systems and Ergodic Theory. More precisely, the main topic of my current research is the study of Lyapunov exponents for linear cocycles.
Lyapunov exponents are quantities that measure the average exponential growth of the iterates of the cocycle along invariant subspaces of the fibers, which are called Oseledets subspaces. An important question in dynamical systems concerns the continuity properties of the Lyapunov exponents, as functions of the input data. This is related to understanding how much Lyapunov exponents vary after a small change in one of its parameters. This allows us to better understand dynamical systems with close parameters. Although in very general contexts they are not continuous functions, it is possible to obtain continuity and modulus of continuity for several classes of linear cocycles that satisfy generic hypotheses. Among them, irreducible cocycles stand out, including the Schrödinger cocycle, much studied in the area of mathematical physics. Therefore, it is an interesting and current problem.
My research, in collaboration with others, focuses on the continuity of the Lyapunov exponents for linear cocycles.
[with Ao Cai, Marcelo Durães and Silvius Klein] Hölder continuity of the Lyapunov exponent for Markov cocycles via Furstenberg's formula, 32 pages, Preprint, 2022. Available on arXiv.
[with Silvius Klein and Xiao-Chuan Liu] Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps, 26 pages, Ergodic Theory and Dynamical Systems (ETDS), 2020. Available here or on arXiv.
[Ph.D. thesis] Markovian, quasiperiodic and mixed dynamical systems, 2023. (PDF)
[Masters thesis] Limit Theorems for Uniquely Ergodic Systems, 2018. (PDF)