We consider the strong chain recurrent set and the generalized recurrent set for continuous maps of compact metric spaces. Recent work by Fathi and Pageault has shown a connection between the two sets, and has led to new results on them. We discuss a structure theorem for transitive/mixing maps, and classify maps that permit explosions in the size of the recurrent sets.
We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.
The topological entropy of a subshift is the exponential growth rate of the number of words of different lengths in its language. For subshifts of entropy zero, finer growth invariants constrain their dynamical properties. In this talk we will survey how the complexity of a subshift affects properties of the ergodic measures it carries. In particular, we will see some recent results (joint with B. Kra) relating the word complexity of a subshift to its set of ergodic measures as well as some applications.
Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of Mercury's eccentricity. However, the mathematical model commonly used to study the problem -- sometimes called the spin-orbit model -- proved not to be entirely convincing, because of the expression used for the tidal torque.
In this talk we will follow the paper titled "Aubry-Mather theory for homeomorphisms", in which it is developed a variational approach to study the dynamics of a homeomorphism on a compact metric space. In particular, they are described orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semidistance. This is work of Albert Fathi and Pierre Pageault.
Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement.
