Georgia Institute of Technology College of Sciences

Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.

This is a preliminary talk for the Workshop "Introduction to Dynamical Systems Methods for Mission Design" that will take place Jan 16-19 in the school of Mathematics. In this talk, we will present the basics of Hamiltonian dynamics and why it is useful. It ishoped that it will be accesible for people with background in undergraduate differential equations who want to participate in the workshop.

A special class of dynamical systems that we will focus on are substitutions. This class of systems provides a variety of ergodic theoretic behavior and is connected to self-similar interval exchange transformations. During this talk we will explore rigidity sequences for these systems. A sequence $\left( n_m \right)$ is a rigidity sequence for the dynamical system $(X,T,\mu)$ if $\mu(T^{n_m}A\cap A)\rightarrow \mu(A)$ for all positive measure sets $A$.

This lecture will discuss the stability of perturbations of integrable Hamiltonian systems. A brief discussion of history, integrability, and the Poincaré nonintegrability theorem will be followed by the proof of the theorem of Kolmogorov on persistence of invariant tori. Time permitting, the problem of small divisors may be briefly discussed. This lecture wIll follow the slides from the Satellite Dynamics and Space Missions 2017 summer school held earlier this semester in Viterbo, Italy.

This presentation is about the results of a paper by L. Bunimovich in 1974. One considers dynamical systems generated by billiards which are perturbations of dispersing billiards. It was shown that such dynamical systems are systems of A. N. Kolmogorov (K-systems), if the perturbation satisfies certain conditions which have an intuitive geometric interpretation.

This presentation is about the results of a paper by Y. Sinai in 1970. Here, I will talk about dynamical systems which resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. It was proved that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such

We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time.

We investigate deterministic superdiusion in nonuniformly hyperbolic system models in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We obtain an explicit formula for the superdiffusion constant in terms of the ne structure that originates in the phase transitions as well as the geometry of the configuration domains of

Dynamical billiards constitute a very natural class of Hamiltonian systems: in 1927 George Birkhoff conjectured that, among all billiards inside smooth planar convex domains, only billiards in ellipses are integrable. In this talk we will prove a version of this conjecture for convex domains that are sufficiently close to an ellipse of small eccentricity. We will also describe some remarkable relation with inverse spectral theory and spectral rigidity of planar convex domains. Our techniques can in fact be

In this seminar,we will explain why and how unpredictable (chaotic) dynamics arises in deterministic systems. Some open problems in dynamical systems, probability, statistical mechanics, optics, (differential) geometry and number theory will be formulated.