DSNT Seminar: Lyapunov exponents of a class of skew-product maps on the two-torus, with expanding base maps

Abstract: Lyapunov exponents of a dynamical system characterize the stability/instability of a system, as they quantify the exponential rate at which the orbits of arbitrarily close points in a system, diverge or converge. Studying the continuity, bounds, and asymptotes of the exponent, gives us valuable insights into the system's hyperbolicity, the existence of synchronisation, invariant measures, attractors, etc. In this talk, we will be interested in skew-product maps of the form F(x,y) = (bx, f_x(y)), where f_x(y) is a circle map on the fiber and b is a large integer. For (well-behaved) linear cocycles, we have the Oseledets decomposition of the space, with respect to its distinct Lyapunov exponents. Whereas for general non-linear maps of the form f_x(y), such a characterisation of the asymptotic behavior of the system is not immediate. Thus, studying the existence of open classes of maps, for which the Lyapunov exponents can be estimated almost everywhere, has been of keen interest. We will discuss one such method and establish a class of circle maps for which the Lyapunov exponents have a uniform negative lower bound. For such systems, on each fiber, almost every point has a stable orbit.

For more information, please contact the organiser Reza Mohammadpour.

This is a seminar in our seminar series on Dynamical Systems and Number Theory (DSNT).