The main topic of my current research is the study of the Lyapunov exponents of a linear cocycle. In ergodic theory, a linear cocycle is a dynamical system on a vector bundle, which preserves the linear bundle structure and induces a measure preserving dynamical system on the base. 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 class of examples of linear cocycles are the ones associated to a discrete, one-dimensional, ergodic Schrödinger operator. Such an operator is the discretized version of a quantum Hamiltonian. Its potential is given by a time-series, that is, it is obtained by sampling an observable along the orbit of an ergodic transformation. The iterates of a linear cocycle may be regarded as a (non-commuting) multiplicative stochastic process.
An interesting and difficult problem is understanding the statistical properties of such processes, under appropriate assumptions. This in turn may have consequences on the behavior of the Lyapunov exponents (e.g. imply positivity or continuity properties) and, in the case of Schrödinger cocycles, on the spectral properties of the corresponding discrete Schrödinger operator.
My research, in collaboration with others, has been concerned with constructing a general theory unifying the connections between the aforementioned topics.
(books, papers, surveys, reports)
- [with Ao Cai and Pedro Duarte] Furstenberg Theory of Mixed Random-Quasiperiodic Cocycles, 46 pages, Preprint, 2022. Available on arXiv.
- [with Pedro Duarte and Mauricio Poletti] Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps, 50 pages, Preprint, 2021. Available on arXiv.
- [with Ao Cai and Pedro Duarte] Mixed random-quasiperiodic cocycles, 29 pages, Preprint, 2021. Available on arXiv.
- [with Xiao-Chuan Liu and Aline Melo] 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.
- [with Pedro Duarte] Large deviations for products of random two dimensional matrices, 67 pages, Commun. Math. Phys. (CMP), 2019. Available here or on arXiv.
- [with Pedro Duarte and Manuel Santos] A random cocycle with non Hölder Lyapunov exponent, 21 pages, Discrete & Continuous Dynamical Systems (DCDS) - A, 2019, 39(8). Available here or here.
- [with Pedro Duarte] Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles, 56 pages, Journal of the European Mathematical Society (JEMS), 2019. Available here or on arXiv.
- [with Pedro Duarte] Continuity of the Lyapunov Exponents of Linear Cocycles, book, 142 pages, Publicações Matemáticas do IMPA, ISBN: 978-85-244-0433-7.
IMPA only prints a limited number of copies of the books in this collection, and ours seems to be sold out now, but here is the link to the pdf file.
- [with Pedro Duarte] Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus, 9 pages, Discrete & Continuous Dynamical Systems (DCDS) - A, 2018, 38 (11), 5379-5387. Available online here.
- Anderson localization for one-frequency quasi-periodic block Jacobi operators, J. Funct. Anal. 273 (2017), no. 3, 1140-1164. Available online here.
- [with Pedro Duarte] Lyapunov Exponents of Linear Cocycles: Continuity via Large Deviations, research monograph, 263 pages, Atlantis Series in Dynamical Systems Vol 3 (2016), ISBN 978-94-6239-123-9, Springer link.
- [with Pedro Duarte] Large Deviation Type Estimates for Iterates of Linear Cocycles, Stochastics and Dynamics, Volume 16, Issue 03 (2016), 54 pages, journal link and pdf file.
- Localization for quasiperiodic Schrödinger operators with multivariable Gevrey potential functions, J. Spectr. Theory 4 (2014), no. 3, 431-484. Available here or on arXiv.
- [with Pedro Duarte] Continuity of the Lyapunov exponents for quasiperiodic cocycles, Comm. Math. Phys. 332 (2014), no. 3, 1113-1166. Available online here.
- [with Pedro Duarte] Positive Lyapunov exponents for higher dimensional quasiperiodic coccycles, Comm. Math. Phys. 332 (2014), no. 1, 189-219. Available online here.
- [with Pedro Duarte] Lyapunov exponents for band lattice quasi-periodic Schrödinger operators, Oberwolfach Report No. 36 / 2013, Mini-Workshop on Direct and Inverse Spectral Theory of Almost Periodic Operators. Available online here.
- Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function, J. Funct. Anal. (2005), no. 2, 255 - 292. Available online here.
- Discrete one-dimensional quasi-periodic Schrödinger operators with non-analytic potentials, Oberwolfach Report No. 51 / 2005, Volume 2, Issue 4, 2005, pp. 2933 - 2978, Mini-workshop on Dynamics of Cocycles and One-Dimensional Spectral Theory. Available online here.
New trends in Lyapunov exponents   (a conference in dynamics and mathematical physics, February 2022)
New trends in Lyapunov exponents   (a virtual conference in dynamics and mathematical physics, July 2020)
Three days in Dynamical Systems   (conference in dynamics at UFRJ, February 2020)
In the shadow of Lyapunov exponents, 2nd edition   (a mini workshop in dynamical systems at PUC-Rio, February 2020)
Produtos aleatórios de matrizes: propriedades estatísticas   (a mini course on Furstenberg theory at PUC-Rio, January 2020)
À sombra de expoentes de Lyapunov   (mini workshop em sistemas dinâmicos na PUC-Rio, janeiro de 2019)
Three days in Dynamical Systems   (conference in dynamics at UFRJ, January 2019)