I am an assistant professor in the Department of Mathematics of the Pontifical Catholic University of Rio de Janeiro (PUC-Rio). I completed my PhD in 2005, at the University of California, Los Angeles (UCLA), where I was part of the Analysis group. My thesis was on spectral theory for discrete, quasi-periodic Schrödinger operators, a mathematical physics topic that uses heavily Fourier analysis tools, and that has strong connections with dynamical systems. I held various postdoctoral positions, of which one, at the University of Lisbon, contributed to my drifting towards dynamical systems. My current research interests are concerned with the study of statistical properties of linear cocycles and their applications to problems in dynamics and mathematical physics.
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 important 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 Pedro Duarte) has been concerned with constructing a general theory unifying the connections between the aforementioned topics.
Selected publications:
- P. Duarte, S. Klein, Large deviations for products of random two dimensional matrices, Commun. Math. Phys. (CMP), 2019
- P. Duarte, S. Klein, Continuity, positivity and simplicity of the Lyapunov exponents for quasi-periodic cocycles, Journal of the European Mathematical Society (JEMS), 2019
- P. Duarte, S. Klein, Lyapunov Exponents of Linear Cocycles: Continuity via Large Deviations, research monograph, Atlantis Series in Dynamical Systems Vol 3 (2016)