(September 11) Dan Rudolph: Generic behavior in positive entropy. The study of generic behavior of measure preserving systems has grown quite rich in recent years. The classical results say a "generic" action is ergodic, of zero entropy, weak mixing, not mixing etc. etc. etc. I will go over what is now known here. My goal though will be to consider something different. For a context, consider the shift invariant Borel probability measures on {0,1}^Z in the weak * topology. For any h strictly between 0 and log2, consider those measures for which the entropy of the shift is >= h. This is a closed set and one can ask what a generic action in this set looks like. The main result I will sketch is that generically a measure in this set gives a Bernoulli action of entropy h. That is to say, there is only one action action in this set up to conjugacy. I will indicate other facts and why they are useful.

(September 18) Yuri Kifer: Some recent advances in averaging. In the study of systems which combine slow and fast motions the averaging principle suggests that a good approximation of the slow motion on long time intervals can be obtained by averaging its parameters in fast variables. A better diffusion approximation suggested by the physicist Hasselmann can be justified provided the fast motion is safficiently fast mixing. When the fast motion depends on the slow one, as it is usually the case, for instance in perturbations of Hamiltonian systems, then the averaging prescription usually works only in some averaged sense if at all. In particular, I shall discuss the case when fast motions are slowly changing Axiom A flows.

(October 2) Genadi Levin: Feigenbaum's dynamics as the nonlinearity grows We study the dynamics of unimodal maps with Feigenbaum's (and other stationary) combinatorics as the order of the critical point increases to infinity. We prove that the limit dynamics exists, as well as there exists a limit of the Hausdorff dimension of the attracting Cantor set. It is bigger than 2/3 but less that 1. Joint works with Greg Swiatek and Feliks Przytycki.

(October 16) Mike Boyle : Entropy on shrinking scales, and the entropy theory of symbolic extensions. This talk reports on papers of myself and Tomasz Downarowicz; Newhouse and Downarowicz; and Downarowicz. The first paper characterizes with functional analysis the existence of a symbolic extension (subshift cover) of a given homeomorphism of a compact metric space with prescribed entropy jumps at invariant measures. The second uses this work and more to give obstructions to the existence of symbolic extensions of C^1 systems (and further results on C^r, with r less than infinity). The last uses this study as a springboard for the creation of a remarkable master entropy invariant for dynamical systems. Tex slides for this talk can be found off my home page.

(November 13) Elena Kosygina: On the homogenization of stochastic Hamilton-Jacobi equation with a vanishing viscosity. We consider a homogenization problem for a stochastic Hamilton-Jacobi equation with a vanishing viscous term. We assume that the Hamiltonian is superlinear and convex with respect to the gradient and stationary and ergodic with respect to the spatial variables. In a very special case the homogenization results can be derived from a work by A. Sznitman on large deviations for quenched path measures for Brownian motion among Poissonian obstacles. The purpose of our work is to obtain homogenization results for more general Hamiltonians by methods, which do not rely on the subadditive ergodic theorem. Joint work with F. Rezakhanlou and S.R.S. Varadhan.

(November 20) Joe Auslander: Weapons of math construction. This is joint work with Ethan Akin and Eli Glasner.
      Topological dynamics is the study of long term or asymptotic properties of families (groups or semigroups) of continuous self maps of topological spaces. An important insight of Robert Ellis is that dynamical properties can be captured by consideration of a suitable compactification of the acting group or semigroup. Examples are the enveloping semigroup and the Stone-Cech compactification.
      We develop the subject by directly considering the (left) action of a compact semigroup S on a compact Hausdorff space X ("Ellis actions"). It is assumed that the maps p --> pq and p --> px from S to S and S to X respectively are continuous; in general there are no other continuity assumptions. This approach was anticipated by Auslander and Furstenberg several years ago in studying product recurrence and distal points.
      This point of view is especially useful in the study of distality concepts. In addition to the usual notion of distality, we consider almost distality (all proximal pairs are asymptotic) and semi distality (there are no non-trivial proximal pairs which are recurrent). There are also relations with Li-Yorke chaos.

(December 4) Anthony Quas : Critical rates in nonconventional ergodic averaging. We consider a number of examples of non-conventional ergodic type averages and describe the maximal rates of divergence of the averages. In particular, we show that averaging along the dyadic sequence of times $2^n$ is in a precise sense the worst possible ergodic average.

(December 8) Werner Ballman : Spectrum and holonomy. I discuss joint work with J.Bruening and G.Carron on an estimate of the smallest eigenvalue of connection Laplacians in terms of the holonomy of the connection.

(December 11) Misha Brin : Weak integrability of invariant distributions. The stable and unstable distributions are uniquely integrable in the sense that the integral manifolds form foliations and any curve tangent to the distribution stays in one leaf. The following weaker integrability property naturally arises in several situations for the central distribution of a partially hyperbolic diffeomorphism. A distribution is weakly integrable if its integral manifolds cover the space but do not necessarily form a foliation.

(January 29) Mike Boyle : Almost isomorphism of countable state Markov shifts. (Joint work with Jerome Buzzi and Ricardo Gomez)
      We introduce an equivalence relation "almost isomorphism" for countable (possibly finite) state Markov shifts. (In the finite state case, these are simply the irreducible shifts of finite type.) Among the countable state Markov shifts, the positive recurrent (PR) shifts are those which have a unique measure of maximal entropy. Within this class, the strongly positive recurrent (SPR) shifts by several criteria are those which resemble the finite state systems: for example, a PR shift is SPR iff w.r.t. the measure of maximal entropy it is exponentially recurrent.
      We prove that finite entropy SPR shifts S and S' are almost isomorphic if and only if they have the same entropy and period. Here for some epsilon > 0, the almost isomorphism induces a shift-commuting bijection F of sets with full measure for every ergodic shift-invariant Borel probability which (i) has full suppport (i.e. is nonvanishing on nonempty open sets) or (ii) has entropy within epsilon of the topological entropy. The sufficiency of criterion (ii) means that S and S' are entropy-conjugate.
      For ergodic measures m,m' satisfying (i) or (ii) and corresponding under F, the map F induces an isomorphism of measure-preserving systems (S,m) --> (S',m') which is finitary (continuous on the complement of a null set) and which for exponentially recurrent measures (such as the measure of maximal entropy) has exponentially fast coding (and in particular, finite expected coding time). This result gives a classification up to entropy-conjugacy for various classes of piecewise smooth systems, on account of earlier work of Jerome Buzzi which reduced their classification to the entropy-conjugate classification of SPR shifts. For example, two topologically mixing piecewise monotonic maps of the interval are entropy-conjugate if and only if they have equal entropy.

(February 5,10,12) Hillel Furstenberg : Multiple Ergodic Averages and Nilpotent Groups
      These lectures will present the background for - and recent results pertaining to - the study of "non-conventional" ergodic averages such as   1/N \sigma f_1(T^n x)f_2(T^2n x)f_3(T^3n x).
      In the first survey lecture we will show the connection between Ramsey theory and "multiple recurrence theorems" in ergodic theory. We will discuss briefly the background in ergodic theory with an emphasis on notions of mixing, and we will summarize recent advances, due notably to B. Host and B. Kra.
      The second and third lectures will discuss in greater detail the structure of ergodic systems, the earlier results available for weakly mixing systems, and the role of special "characteristic factors". Finally we shall explain the special role played by nilpotent groups and their homogeneous spaces, introducing in this connection something that might be called "ergodic geometry".

(February 26) Marco Lenci : Recurrence, ergodicity and their typicality in (aperiodic) Lorentz gases
      A Lorentz gas is here defined as the billiard system in the complement of a disjoint collection of strictly convex bounded sets in the plane. No periodicity condition is required. Subject to reasonable geometrical hypotheses, this infinite-measure dynamical system is ergodic if it is recurrent. We show this fact, give examples of recurrent gases and discuss the question of how typically a Lorentz gas is recurrent, and therefore ergodic.

(March 4) Dmitry Dolgopyat : Variations on Anosov Theorem
      Anosov's Thesis written about 40 years ago gave the first justification of the averaging method in a quite general setting. I describe various extensions of Anosov's result.

(March 11) Bruce Kitchens : Dynamical rigidity for Z^d-actions
      An algebraic Z^d-action consists of a compact topological group together with d commuting group automorphisms. Haar measure is preserved by the automorphisms. A dynamical system is rigid if any map from another dynamical system onto it which preserves Haar measure and which commutes with the actions is forced to be an algebraic map. This is the case for certain algebraic Z^d actions when d > 1. I will describe how this comes about in a zero entropy setting and how the algebra of the group forces this property.

(April 1) E. A. Robinson Realizing model set dynamical systems with a given spectrum
      Model sets were first defined by Y. Meyer in the 1970's in the context of harmonic analysis and number theoory, but became a staple in the theory of quasicrystals in the 1990's. Model sets are special uniformly discrete and uniformly dense subsets of R^d that are typically aperiodic. The points in the model set represent the atoms of the quasicrystal. Model sets all have pure point diffraction spectrum. Mathematically this means they have a discrete Fourier transfrom. Physically this means they have an X-ray diffraction pattern with sharp spots. Any model set defines a minimal dynamical system (on its translational orbit closure), that is always an almost 1:1 extension of a Kronecker system (an action by rotations on a LCA group). Moreover, the dynamical point spectrum of the kronecker system is the same as the diffraction spectrum of the model set. A famous result says that any dense subgroup of R^d can be the spectrum of an R^d Kronecker system. We examine the question of whether the same result is true for the diffraction spectra of model sets.

(April 8) Carlangelo Liverani : Banach spaces adapted to anosov systems
      I present Banach spaces of distribution $B_r$ with the property that the transfer operator associated to an Anosov maps has spectral radius one and essential spectral radius $\lambda^{r}$. This allows to easily treat issues of stability (both stochastic and non) also in higher smoothness and may allow to study of the properties of the dynamical zeta function.

(April 13) Ilya Goldsheid : Towards a constructive approach to Lyapunov exponents.
      The desire for constructive estimates of Lyapunov exponents has been in place since a very long time ago - perhaps since the publication of the famous Fursatenberg's theorem (1963) stating that the top Lyapunov exponent of a product of random matrices is strictly positive. These estimates are essential for a number of applications. I shall explain how they can be obtained and discuss some of the applications. I shall also explain how these results lead to a proof of the Furstenberg's theorem.

(April 22) Dmitry Dolgopyat : Regularity of transport coefficients for Lorenz gas
      I review known results about the existence of transport coefficients in billiard systems and then discuss how these coefficients depend on parameters. This is a joint work with N. Chernov

(April 29) Roger Jones : Variation inequalities in ergodic theory
      For the abstract, click here.