Activities This Week

Let T be a bounded linear operator on a Banach space X; the elements of (I − T)X are called coboundaries. For two commuting operators T and S, elements of (I − T)X ∩ (I − S)X are called joint coboundaries, and those of (I − T)(I − S)X are double coboundaries. By commutativity, double coboundaries are joint ones. Are there any other? Let θ and τ be commuting invertible measure preserving transformations of (Ω, Σ, m), with corresponding unitary operators induced on L2(m). We prove the existence of a joint coboundary g ∈ (I − U)L2 ∩ (I − V )L2 which is not in (I − U)(I − V )L2. For the proof, let E be the spectral measure on T 2 obtained by Stone’s spectral theorem. Joint and double coboundaries are characterized using E, and properties of the maximal spectral type of E, together with a result of Foia³ on multiplicative spectral measures acting on L2, are used to show the existence of the required function.

The billiards in the plane with periodic obstacles are dynamical systems with a simple description but intricate features in their behavior. A specific example, introduced by Paul and Tatania Ehrenfest in 1912, is the so-called “wind-tree” model, where a ball reduced to a point moves on the plane and collides with parallel rectangular scatters according to the usual law of geometric optics.

Natural questions are: does the ball return close to its starting point (recurrence), how fast the ball goes far from it? (diffusion), what is the set of scatters reached by the ball?

These billiards can be modeled as dynamical systems with an infinite invariant measure. The position of the particle can be viewed as a stationary random walk, sum of a stationary sequence of random variables with values in $R^2$, analogous to the classical random walks. For the billiard the increments are the displacement vectors between two collisions, while for the classical random walks the increments are independent random variables.

In the talk, after some general facts about systems with infinite invariant measure, the notions of recurrence and growth (or diffusion) of a stationary random walk will be illustrated by examples, in particular the “wind-tree” model.