(September 26) Steve Kalikow: When are random graphs connected? - Joint work with Benjamin Weiss.

(October 01) Hillel Furstenberg: Self-similarity of fractal measures and measure-valued stationary processes - The self similarity property of certain fractals in Euclidean space appears as one "zooms down" to a point of the fractal with the resulting sequence of pictures repeating periodically. A more common phenomenon would be to obtain a "recurrently" repeating sequence, of the type familiar from typical sample sequences of stationary time series. This is best studied by considering measures on Euclidean space rather than "pictures", and see what happens as one "zooms in" on points and rescales the measure. We arrive at a notion of "recurrent measures"; we'll give examples of these, and show how the most general ones are obtained.

(October 3) Hillel Furstenberg: What's New in the Theory of Non-Conventional Ergodic Averages? - This will be a report on recent work of Host and Kra which explains the role of nilsystems (dynamical systems on nilmanifolds) in evaluating non-conventional ergodic averages for general systems, such as the average of f(T^n(x))g(T^2n(x)h(T^3n(x))

for n between 1 and infinity.

(October 10) Mike Sullivan: Flow equivalence of skew-products of irreducible shifts of finite type - Work in Progress. Joint with Mike Boyle.

Let G be a finite group, and F be a function from the edge set of a given SFT into G. This can be thought of as a skew-product system or a G-weighted SFT. A G-weighted SFT is determined by a square matrix over the semi-group ring Z_+G. Strong shift equivalence (SSE) over Z_+G is defined and is equivalent to the existence of a conjugacy that respects the G-weighting on cycles (defined by multiplication along edge labels in G) in the irreducible case. But G-weighted SSE is even harder to determine than the usual SSE, so we pass to a suitable analog of flow equivalence. We are in the process of proving(!) that G-weighted flow equivalence is generated by similarity over SL(ZG) (we use square matrices indexed by the natural numbers) for nontrivial irreducible systems. This should vacillate the computing of invariants for specific groups G. In particular, when G=Z/2Z we believe a complete computable classification can be found. This case is called twistwise flow equivalence and has been applied to understand the twisting in the local stable manifolds of basic saddle sets in Smale flows.

(December 12) Mike Sullivan: A Linking Homorphism for Minimal Sets. - Joint with Alex Clar

(March 13) Jim Wiseman : Sofic shifts from signed matrices - We consider a method for assigning symbolic dynamics (a sofic shift) to a square matrix with arbitrary real entries (i.e., not necessarily just nonnegative integers) by associating to it a directed graph with some vertices labeled 1 and the rest 2 (in applications the choice of labeling should be natural). We can obtain an estimate for the topological entropy of this sofic shift by comparing the characteristic polynomial of the original matrix to those of the matrices for the restrictions of the shifts to each piece (1 and 2). Our main application is to the use of the Conley index to detect symbolic dynamics in isolated invariant sets.