(September 8) Mike Boyle : Symbolic extension entropy: C^r examples, products and flows
(This is joint work with Tomasz Downarowicz.)
Adapting techniques of Misiurewicz, for r in [1,infinity) we give an explicit construction of C^r maps with positive residual entropy (i.e., with symbolic extension entropy greater than the topological entropy). We also establish the behavior of symbolic extension entropy with respect to joinings, fiber products, powers and flows.

(September 15) Michael Schraudner : Algebraic properties of automorphism groups of countable state Markov shifts
We study the algebraic properties of automorphism groups of two sided, transitive, countable state Markov shifts together with the dynamics of those groups on the shiftspace, on periodic orbits and on the 1-point-compactification of the shiftspace. Comparing those properties to results about the automorphism groups of SFTs, we find some similarities as well as some differences. We present a complete solution to the cardinality-question of the automorphism group for locally compact and non locally compact, countable state Markov shifts, shed some light on its huge subgroup structure and prove the analogue of Ryan's theorem about the center of the automorphism group in the non compact setting. Moreover we characterize the 1-point-compactification of locally compact, countable state Markov shifts, whose automorphism groups are countable. Those are conjugate to synchronized systems on doubly transitive points. Finally we show the existence of a class of locally compact, countable state Markov shifts whose automorphism groups split into a direct sum of two groups; one being the infinite cyclic group generated by the shift map, the other being a countably infinite, centerless group, which can be specified to some extent.

(September 22) V.S. Prasad : Multitowers, Conjugacy and Coding
We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation $\sigma$ of a Lebesgue probability space $(X, \mathcal{A}, \mu) $ and show that each theorem is a corollary of any other. One theorem asserts that the conjugates of $\sigma$ are dense in various topologies on the space of automorphisms. The other two results concern the existence of a partition $\mathcal{P} = \{ P_i \}$ with special properties. The first partition result is Alpern's generalization of the Rokhlin Lemma, the so-called Multitower Rokhlin theorem stating that the space can be partitioned into denumerably many $\sigma$-columns (the $i$'th column given by $P_i$) with the measures of the columns prescribed in advance (this has proved useful in dynamics on the Cantor set); the second partition result is a coding result which asserts that any mixing Markov chain $\mathbf{P}=\left(p(i,j)\right)$ can be represented by a partition $\mathcal{P}= \{P_i\}$ and $\sigma$ so that $p(i,j) = \mu(\sigma (P_i) \cap P_j)/\mu(P_i)$ (the latter result has been useful in rotational representations of Markov Chains). This is joint work with S. Alpern at LSE.

(October 6) Yakov Pesin : Existence of hyperbolic Bernoulli flows on any manifold of dimension > 2
I will discuss a recent affirmative solution of the long-standing problem of constructing a volume-preserving Bernoulli flow with nonzero Lyapunov exponents on any smooth compact manifold of dimension greater than two.

(October 13) Christian Bonatti : Existence of homoclinic intersections
Homoclinic intersections allways implies a chaotic behavior of the dynamics. Recently, Pujals and Sambarino in dimension 2, myself with Gan and Wen in dimension 3 and finally Crovisier in any dimension proved that any diffeomorphism (on any compact manifold) may be approached (in the C1-topology) by diffeomorphisms which are either very simple (Morse-Smale) or present homoclinic intersection. After introducing the main notion, I will try to present the main ideas leading to this result.

(October 20) Steve Zelditch : Quantum ergodicity, dynamical zeta functions and Patterson-Sullivan distributions
Quantum ergodicity concerns eigenfunctions of the Laplacian, in particular the Wigner distributions in phase space associated to them. The purpose of my talk is to relate these Wigner distributions to Patterson-Sullivan distributions in the case of a hyperbolic surface. P-S distributions are residues of purely classical dynamical zeta functions. Thus, on hyperbolic surfaces, quantum ergodicity can be reduced to a purely classical dynamical problem. This is joint work with Nalini Anantharaman.

(October 27) Rafael de la Llave : Recent progress in geometric methods for instability in Hamiltonian systems
It is well known -- and presumably true -- that many Hamiltonian systems subject to perturbations become unstable. We want to discuss several different mechanisms for such instability. The mechanisms are based on identifying invariant objects that provide with roads for the trajectories to scape. These mechanisms have different quantitative properties. We will discuss how to identify these mechanisms in concrete systems.

(November 3) Artur Lopes : Maximizing measures, large deviations at temperature zero and eigenvalues of the Laplacian in negative curvature
Click here .

(November 10) Amie Wilkinson : C^1 genericity of trivial centralizers
(This is joint work with Sylvain Crovisier and Christian Bonatti.)
The *centralizer* C(f) of a diffeomorphism f is the set of all diffeomorphisms that commute with f. C(f) is naturally viewed as the group of symmetries of f; for example, if f embeds in a flow, then C(f) contains that flow, which is isomorphic to the real numbers. We say that f has *trivial centralizer* if C(f) is precisely the group generated by f; that is, f commutes only with its iterates f^n, n in Z. Smale has asked whether any diffeomorphism can be approximated by one with trivial centralizer. In this talk I will discuss answers to this and related questions, in the C^1 topology on diffeomorphisms.

(December 1) Vadim Kaloshin : Nonlocal Arnold diffusion for the Restricted Planar Circular 3 Body Problem
The Restricted Planar Circular 3 Body Problem (RPC3BP) is the simplest nonintegrable 3 body problem. Usually it is viewed as a model for planar either Sun-Jupiter-Asteriod or Sun-Earth-Earth Satellite system. Stability v.s. instability of such a system is one of long standing problems. We consider the first model. Using Aubry-Mather theory, Mather variational method, and numerical analysis, we managed to prove existence of rich variety of unstable motions. For example, an Asteriod could have a nearly elliptic orbit of say eccenticity 0.76 in the past and escape to infinity along nearly parabolic orbit of eccentricity more than 1. These motions could be interpreted as Arnold diffusion for this system. This is a joint work with T. Nguyen and D. Pavlov.

(December 8) Victor Sirvent: Space filling curves and geodesic laminations
In this talk we shall associate space filling curves to connected fractals, obtained as the fixed point of an iterated function systems (IFS) satisfying the common point property and other conditions. These curves are H\"older continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.

(February 2) Michael Keane : Coefficient dynamics
In this lecture, I describe an old method in a new setting which may be useful for the study of symbolic sequences obtained by expanding numbers either in continued fractions or in bases. After an introduction to the method, I intend to sketch an earlier result of mine which describes a possible way in which Gauss discovered the statistical distribution of partial quotients of continued fractions, some two hundred years ago. Then, using these ideas, I present what seems to be a new proof of an old theorem of Lagrange, stating that quadratic algebraic numbers have eventually periodic continued fraction expansions. Finally, I would like to discuss the conjecture that there are no irrational algebraic numbers belonging to the classical Cantor set, and present a new conjecture using dynamics which, if true, would lead to a partial solution of this problem.

(February 9) Joe Auslander : Non-compact dynamical systems
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(February 16) Michael Keane : Finitary orbit equivalence
Last year, in an article with T. Hamachi to appear in the Bulletin of the LMS, we showed that the binary and ternary odometers are orbit equivalent by exhibiting a finitary isomorphism of their corresponding measure spaces taking full orbits to full orbits. Now we have extended this result in a number of ways:

1. My student Mrinal Roychowdhury has shown that any two m-adic odometers (also with variable bases m) are finitarily orbit equivalent. Subsequently he has also shown that irrational circle rotations satisfy the same theorem, in their usual topology.
2. The three of us have shown that any two measured Bratteli diagrams are finitarily orbit equivalent (in the sense that the corresponding equivalence relations are preserved under the finitary mapping). Together with a theorem of Vershik's this shows that any ergodic measure preserving transformation has a topological model which is finitarily orbit equivalent to the binary odometer, extending Dye's theorem and providing an alternate proof.
3. With T. Hamachi and Yuasa we have now shown, using these methods, how to complete the suggestion of Glasner and Weiss from 1995 to reprove the theorem of Giordano, Putnam, and Skau on topological orbit equivalence of Cantor minimal systems. This article is still in preparation, but a preliminary version is available.

(February 23) Giovanni Forni Weak mixing for Interval Exchange Transformations
We will describe our joint work with A. Avila on the weak mixing property for almost all interval exchange transformations and for almost all directional flows on almsot all translation surface. The proof is based on previous work on Lyapunov exponents of a renormalization cocycle introduced and studied by Rauzy, Veech, Zorich, Kontsevich among others, and on a parameter elimination scheme.

(March 8) Ethan Akin Which Bernoulli measures are good?
A measure m on Cantor Space is a 'good measure' when it satisfies a simple homogeneity property: If U and V are clopen sets with m(U) < m(V) then there is a clopen set W contained in V with m(U) = m(W). Despite the apparent mildness this leads to strong results. The good measures are exactly the Jewett-Krieger measures, i.e. the invariant measures of uniquely ergodic minimal homeomorphisms on Cantor Space. Bernoulli measures, which arise naturally from the topologically transitive but very nonminimal shift homeomorphisms, are not the natural place to look for good measures. Nonetheless Dan Mauldin and his colleagues have produced a lovely characterization of the good ones among the Bernoulli measures for a 2 letter alphabet. I will describe some partial results and open questions for the n letter alphbet case.

(March 8) Eli Glasner The local variational principle
The classical variational principle asserts that the topological entropy of a compact dynamical system (X,T), equals the supremum over the measure entropies h_\mu where \mu ranges over the set  of T- invariant probability measures on X. However, given an open cover U, the classical VP says little about the relation of the topological entropy of U to measure theoretical entropy. I will describe a series of recent works by several authors which develop a theory of local variational principle and the related theory of entropy pairs.

(March 15) Jack Feldman Multiple ratio ergodic theorems
In 1985 Krengel and Brunel gave a counterexample to the possibility of proving a two-dimensional version of Hurewicz's Ratio Ergodic Theorem (summing over a square in the positive quadrant). However about a year ago it was shown that if one sums over symmetric squares or hypercubes, instead, such a theorem may be proven (in d dimensions). In the present talk, after a review of this, we investigate the possibility of successfully summing over other regions, such as Euclidean balls.

(March 30) Carlangelo Liverani Dynamical systems from a functional analytic point of view: a brief overview
The last years have witnessed the development of new approaches to study the statistical properties of dynamical systems based on the direct study of the transfer operator together with, eventually, a minimal coding of the dynamics. I will review some of my results in this context trying to emphasize the breadth of the approach.

(April 6) Bill Goldman Proper Affine Actions of Free Groups
In 1977 Milnor asked whether a free group could act properly by affine transformations on Euclidean space. In the early 1980's Margulis showed the existence of proper affine actions on Euclidean 3-space. This talk will describe the construction and classification of proper 3-dimensional affine actions of finitely generated groups.

(April 13) Dmitry Dolgopyat Non-compact dynamical systems
We describe methods to establish recurrence of infinite measure dynamical systems

(April 19) Dan Mauldin Divergent averages along squares
Answering a question of Bourgain, we indicate the construction of an L^1 function f and an ergodic system such that the averages of f along squares diverge almost surely.

(April 28) Mark Levi Arnold diffusion in nearly flat metrics on tori
In this joint work with Vadim Kaloshin we construct a metric on the 3-torus, arbitrarily close to flat metric, in which Arnold diffusion is particularly transparent.