Since the early 1970's, it has been known in both, the mathematical physics and in the physics communities, that propagation of information in quantum spin chains cannot exceed the so-called Lieb-Robinson bound (effectively providing the quantum analog of the light cone from the relativity theory). Typically these bounds depend on the parameters of the model (interaction strength, external field). The recent Hamza-Sims-Stolz result demonstrates exponential localization (a la Anderson localization) of information propagation in most spin chains (in the sense of a given probability distribution with respect to which interaction and external field couplings are drawn). A natural question arises: what can be said about lower bounds on propagation of information in spin crystals (i.e. the case far from the one in which localization is expected), as well as in the intermediate case--the spin quasicrystals. This problem can be reduced to solving a linear ODE given by a Hermitian matrix, the solutions of which live on finite-dimensional complex spheres.

In this talk we shall discuss the history, give a general overview of the field, reduce the problem to an ODE problem as mentioned above, and look at some open problems. We shall also present some numerical computations with animations.

There is known a lot of information about classical or standard shadowing. It is also often called a pseudo-orbit tracing property (POTP). Let M be a closed Riemannian manifold. Dieomorphism f : M \to M is said to have POTP if for a given accuracy any pseudotrajectory with errors small enough can be approximated (shadowed) by an exact trajectory. A similar denition can be given for flows.

Most results about this property prove that it is present in certain hyperbolic situations. Quite surprisingly, recently it has been proven that a quantitative version of it is in face equivalent to hyperbolicity (structural stability).

There is also a notion of inverse shadowing that is a kind of a converse to the notion of classical shadowing. Dynamical system is said to have inverse shadowing property if for any (exact) trajectory there exists a pseudotrajectory from a special class that is uniformly close to the original exact one.

I will describe a quantitative (Lipschitz) version of this property and why it is equivalent to structural stability both for dieomorphisms and for flows.

Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation among all possible averages. We consider
1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function,
2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and
3) oscillatory behavior of these norm approximations.