(September 7) Hillel Furstenberg : NOTE: FALL 2000: Fractals, ergodic theory and geometric Ramsey theory - Ramsey Theory deals with the phenomenon that large subsets of certain structures (Euclidean space, complete graphs, integers, etc.) necessarily contain "miniature models" of the structure. Some of these phenomena are related to notions of recurrence in ergodic theory, and this has been useful, e.g., in studying sets of integers of positive density. In the present lectures we will apply a variant of these ideas to the geometry of fractals of positive Hausdorff dimension, and, more generally, to large finite subsets of Euclidean space corresponding to "positive box-dimension". One of the tools in this analysis involves stochastic processes taking values in the space of measures on Euclidean space, where the passage of time corresponds to a rescaling of the measure. A third lecture is planned (not in the framework of the dynamics seminar) in which we will pursue this idea.

(September 21) Martin Sambarino: Density of Hyperbolicity and Homoclinic Bifurcation - We'll present some results regarding global dynamics on surfaces, existence of homoclinic orbits and variation of topological entropy. Moreover, we'll show how these results follow from the study of systems having a dominated decomposition.

(September 28) Jose Alves: Nonuniformly expanding dynamics: Stability from a probabilistic viewpoint - We present some recent developments on the theory of smooth dynamical systems exhibiting nonuniformly expanding behavior. In particular we show that these systems have a finite number of SRB measures whose basins cover the whole ambient space, and under some conditions on the rate of expansion their dynamics is statistically stable.

(October 5) Susan Williams : Dynamical Coloring of Knots - A classical elementary invariant of knots is the group of Fox $n$-colorings. Generalizing the Fox coloring theory, we associate to a link of $d$ components and compact group $\Sigma$ a $Z^d$ shift of finite type. The conjugacy class of this shift is a link invariant for every $\Sigma$. Its periodic point structure gives us topological information about the finite abelian covers of $S^3$ branched over the link. This work was done jointly with Dan Silver.

(October 12) Dan Silver: Knots, Links and Lehmer's Question - The Mahler measure of a polynomial is the geometric mean of its absolute value on the circle |z|=1. In 1933 D.H. Lehmer displayed a polynomial with integer coefficients and Mahler measure approxomately equal to 1.17. No polynomial with integer coefficients has yet been found with Mahler measure less than this value but greater than 1, despite extensive computer-assisted searches. Lehmer's Question asks whether or not such a polynomial exists. A knot or link gives rise to an algebraic dynamical system with topological entropy equal to the Mahler measure of its Alexander polynomial. In fact, Lehmer's polynomial appears in this context. We will discuss a relationship between knot theory and Lehmer's Question.

(October 19) Paul Shields : Some new sample path concepts and their implications - If a process is ergodic and k is fixed, then by the ergodic theorem the observed distribution of overlapping k-blocks in an n-sequence converges to the true distribution almost surely as n goes to infinity. This talk will focus on what can be said when k=k(n) is allowed to grow with n, but no faster than (log n)/entropy. Much has been done in the case when the process has very strong mixing properties, e. g., the distance between the observed k(n)-distribution and the true k(n)-distribution goes to 0 almost surely. Furthermore, given any rate of growth on k(n), there are ergodic processes for which this is not true, and this appears to be all that can be said. Nevertheless, I will state some positive theorems and suggest a kind of local ergodicity property that holds for all known examples and if it held in general would have lots of nice consequences.

(November 1) Mark Levi: Stabilty of the inverted pendulum by a topological argument. - Consider a pendulum - a rigid rod pivoting around a point O without friction. If the pivot O is forced to vibrate vertically with high enough frequency, the rod's upside-down position becomes stable. This fascinating phenomenon, known since 1908, has been addressed in dozens of papers. We will give a simple topological explanation of stabilization by vibration. This explanation is based on visualizing the set of all 2 by 2 matrices with determinant 1 (known as SL(2,R)). A physical demonstration will be given.

(November 2) Mark Levi: Periodic motions of charged particles in magnetic fields. - We describe a recent result answering a question posed by Arnold on the existence of periodic motions of a charged particle on the two-torus in the presence of magnetic field perpendicular to the torus. Along the way we give a new proof of Moser's density uniformization theorem.

(November 9) Jose Alves: Stochastic stability of nonuniformly expanding maps - We give both sufficient conditions and necessary conditions for the stochastic stability of nonuniformly expanding maps. We also show that the number of probability measures describing the statistical behavior of random orbits is bounded by the number of SRB measures if the noise level is small enough.

(November 16) Chaim Goodman-Strauss: Aperiodic tilings in the hyperbolic plane and symbolic dynamics - A set of tiles is "(strongly) aperiodic" iff they do admit a tiling, but somehow admit no tiling with an infinite cyclic symmetry (i.e. a period).

Such tiles were first found (in the Euclidean plane) over thirty years ago, but even today seem paradoxical. Somehow the tiles force non-periodic structures to emerge over vast, arbitrarily large distances. The existence of such sets of tiles is a direct consequence of the undecidabilty of the "Domino Problem". This in turn follows from the fact that tiling can serve as a universal model of computation.

There are many good reasons to wonder if such aperiodic sets of tiles can exist in the hyperbolic plane. In particular, it is still open whether the Domino Problem is decidable in this setting. Moreover, as we show, every previously known tiling in H^2 corresponds nicely to "substitution system"; these are in turn fairly straightforward gadgets.

But a priori, tilings in H^2 correspond to slightly more general objects, substitutions on regular languages. Mysteriously, these seem to have the magic, that is, the combinatorial complexity needed for universal computation. Though that result remains elusive, this has led, at last, to a strongly aperiodic set of tiles in the hyperbolic plane.

(November 30) Jim Yorke: Smooth embedding of non-smooth sets - Takens, Ruelle, and Eckmann launched an investigation of images of attractors of dynamical systems. If a compact set A is an attractor in R^n and g:R^n -> R^m is a generic smooth map, and if m < n, how do A and g(A) compare? By making physical measurements, physicists in effect are examining a set g(A) and would like to recover as many properties of A by examining g(A), hence the need for understanding how they compare. If A is a manifold, the problem is relatively easy, but that is rarely the case if the dynamics are chaotic. When A is chaotic, the properties studied include the dimension of A or the Lyapunov exponents.

(November 30) Enrique Pujals: Robust transitivity and partial hyperbolicity. - The idea would be to explain the relation between robust transitivity and partial hyperbolicity, and we could try to discuss necessary and sufficient conditions to characterize systems which are robust transitive.

(December 7) Bill Goldman: Automorphisms of cubic surfaces - For a real parameter t, consider the surfaces S(t) in R^3 defined by the cubic polynomial

k(x,y,z) := x^2 + y^2 + z^2 - xyz -2.

Its group G of automorphisms is an infinite discrete group, containing polynomial maps of unbounded degree. It is commensurable to the modular group SL(2,Z), and is generated by quadratic reflections. Furthermore there is a Poisson structure on R^3 in which G is area-preserving on each level surface S(t).

For t < -2, the action is properly discontinuous, whereas for 2 < t < 18, the action is ergodic. For other values of t, the behavior is mixed. The action is closely related to Hamiltonian flows.

These surfaces arise from moduli spaces over Riemann surfaces, and the cubic polynomial is related to the structure of Markoff triples.

(February 8) Marcelo Viana: Lyapunov exponents: zero or non-zero ? -

I'd speak on very recent results of Bochi and myself, and Bonatti, Gomez-Mont, and myself, about genericity of zero and of non-zero Lyapunov exponents, in various situations.

(February 22) Misha Brin: Partial hyperbolicity - Stable and unstable foliations, accessibility, topological transitivity, ergodicity.

(March 8) Mike Boyle: The residual entropy of a dynamical system - See the abstract and introduction of the preprint "Residual entropy, conditional entropy and subshift covers" with D. Fiebig and U. Fiebig, in postscript , pdf , or dvi format.

(March 15) Francois Ledrappier: Ergodic properties of linear actions of 2 x 2 matrices - Let $\Gamma $ be a discrete cocompact group of $2 \times 2$ real matrices and consider the natural linear action on the plane. We'll recall ergodic properties of that action, and discuss their extensions to $2 \times 2$ matrices on other number fields. (This is a joint work with Mark Pollicott).

(March 29) Michiko Yuri: Statistical properties of weak Gibbs measures - In this talk, we study mathematical models which show both phase transition and subexponential decay of correlations. For this purpose, we introduce a new class of potentials and a weak notion of the Gibbs property (in the sense of Bowen) for piecewise $ C^0 $-invertible Markov systems. We present a new method for the construction of conformal measures associated to the potentials which satisfy the weak Gibbs property and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits force the decay of correlations to be slow and cause failure of uniqueness of equilibrium states.

(April 5) Karen Ball: Equivalence of random walks on a random scenery - A random walk on a random scenery is a measurable transformation constructed by taking a skew product of the Bernoulli 2-shift with an invertible transformation T (or a flow T_t). Heicklen, Hoffman and Rudolph have proven that for a class of these examples, the T,T-inverse maps, the entropy of T is an isomorphism invariant. This talk will describe this result and efforts to extend it to other classes of random walks on random sceneries.

(April 12) E. Arthur (Robbie) Robinson: Tilings associated with some non-Pisot matrices and explicit construction of Markov partitions -

Let A be a dXd integer matrix with a 2-dimensional expanding subspace W^+. Consider substitutions \theta on d symbols having A as their "structure matrix". In particular, we allow generalized substitutions in which entries may look like a^(-1). These are essentially free group endomorphisms. We think of A as the abelianization of \theta and each \theta (there are many) as a "non-abelianization" of A.

Let p_1, ..., p_d be the projections of the standard basis vectors to W^+. In a natural way each substitution \theta defines a piecwise linear boundary curve made from these vectors. The goal is to try to tile the interior of these curves by the C(d,2) rhombic prototiles having these vectors as edges. If a \theta is found such that this is possible then we get a family of self-similar tilings of the plane W^+.

In the case this works for both A and A^(-1) (so d=4) we get an explicit Markov partition for A. Of course, it usually has a fractal boundary. In this case both W^+ and W^- are two dimensional, and we refer to A as a non-Pisot matrix.

A sufficient condition for success of the program for a matrix A is the non-negativity of an associated C(d,2)XC(d,2) matrix which we denote by A^*. Sufficient conditions for this can be given. In particular, it suffices that A be symmetric. Thus we obtain an explicitly constructed Markov partition for 4X4 symmetric matrices satisfying the non-Pisot condition.

This talk is joint work with Maki Furukado & Shunji Ito.

(April 26) Valentyn Golodets: The spectrum of completely positive entropy actions and the Rudolph-Weiss theory - We prove that an ergodic free action of a countable discrete amenable group with completely positive entropy has countable Lebesgue spectrum. It is a generalization of the Rohlin-Sinai result for Z. Our approach is based on the recent Rudolph-Weiss result on the equality of conditional entropies for actions of countable amenable groups with the same orbits.

(May 3) Valentyn Golodets: The spectrum of completely positive entropy actions and the Rudolph-We iss theory II - This is a continuation of last week's talk.

(August 30) Tomasz Downarowicz: Entropy Properties in Non-uniquely Ergodic Systems - In a non-uniquely ergodic system (action of a single homeomorphism T) the seemingly most complete information about entropy is in the entropy function (assigning to each invariant measure the entropy of T with respect to this measure). However, for many aspects like asymptotic h-expansiveness, existence of a subshift cover, etc., we need a more subtle tool, namely we need to know how the entropy function is approximated by certain relative entropy functions. We will present some recent results concerning the entropy function, relative entropy functions, relative variational principles, existence of subshift covers, criteria for asymptotic h-expansiveness, and the like. We will propose a new topological invariant (which we call "entropy structure") carrying complete information about the above discussed entropy properties of the system.