(October 7) Yakov Pesin: Stable Ergodicity of Partially Hyperbolic Diffeomorphisms: The Dissipative Case — I describe two "competing" methods to show that a given partially hyperbolic diffeomorphism is stably ergodic (i.e., it is ergodic along with any of its sufficiently small perturbations). One of them relates the problem to the global estimates of the action of the system along its central direction while another one deals with a more delicate estimates using Lyapunov exponents in the central direction.
(October 28) Shmuel Friedland: The pressure, densities and first order phase transitions associated with multidimensional SOFT — We discuss theoretical and computational properties of the pressure function for subshifts of finite type on the integer lattice $Z^d$, multidimensional SOFT, which are called Potts models in mathematical physics. We show that the pressure is Lipschitz and convex. We use the properties of convex functions to show rigorously that the phase transition of the first order correspond exactly to the points where the pressure is not differentiable. (We avoid the use of Gibbs measures.)
We give computable upper and lower bounds for the pressure, which can be arbitrary close the values of the pressure given a sufficient computational power. We apply our numerical methods to confirm Baxter's heuristic computations for two dimensional monomer-dimer model, and to compute the pressure and the density entropy as functions of two variables for the two dimensional monomer-dimer model. A complete version of the paper on this subject is available at http://arxiv.org/abs/0906.5176 This paper is a continuation of S. Friedland and U.N. Peled, Theory of Computation of Multidimensional Entropy with an Application to the Monomer-Dimer Problem, Advances of Applied Math. 34(2005), 486-522, (top cited article 2005-2010 of this journal). An old variant of this talk is available at http://www2.math.uic.edu/$\sim$friedlan/preshusem12.06.pdf
(November 4) Zhenqi Wang: Local rigidity of generic partially hyperbolic abelian algebraic higher-rank actions — We prove the local differentiable rigidity of generic partially hyperbolic abelian algebraic higher-rank actions on compact homogeneous spaces obtained from various simple groups. The conclusions are based on the geometric approach by Katok-Damjanovic and a progress towards computations of the generating relations in these groups.
(November 11) Vaughn Climenhaga : Thermodynamics for non-uniformly mixing systems: factors of beta-shifts are intrinsically ergodic — Thermodynamic formalism begins by realising the topological entropy of a topological dynamical system as the supremum of the measure-theoretic entropies, taken over all invariant measures. If there exists a unique measure achieving this supremum, the system is called intrinsically ergodic.
It is well known that intrinsic ergodicity holds for expansive systems satisfying a uniform mixing property (Markov, sofic, specification). We introduce a new technique for dealing with systems which are not uniformly mixing, and give verifiable criteria for intrinsic ergodicity.
As a corollary, we show that every subshift factor of a beta-shift is intrinsically ergodic, which answers an open question of Klaus Thomsen.
(November 18) Marcel Guardia: Analytic properties of one and a half degrees of freedom Hamiltonian Systems and exponentially small splitting of separatrices —
The exponentially small splitting of separatrices appears naturally in analytic integrable Hamiltonian Systems with a fast periodic perturbation. In this setting Melnikov theory fails, in general, to predict correctly the distance between the invariant manifolds due to the exponential smallness. Moreover, this distance is extremely sensitive to the analyticity properties of the Hamiltonian System. In this talk we will consider first Hamiltonian Systems which are polynomial or trigonometric polynomial in their variables and we will explain how the size of the splitting depends on what was called by T. M. Seara and A. Delshams the order of the perturbation. Then, we will consider an example of meromorphic Hamiltonian System and we will see that the size of the splitting strongly depends on its width of analyticity. This is joint work with I. Baldoma, E. Fontich and T. M. Seara
(December 2) Jinxin Xue: Continuous averaging and the Nekhoroshev theorem — In this work, Treschev's continous averaging has been developed to the simultaneous Diopantine approximation case. When applied to the Nekhoroshev theorem, it gives a sharp estimate of the stability constant C_2.
(February 24) Matt Bainbridge: The classification problem for Teichmuller curves — A Teichmuller curve is a totally geodesic curve in the moduli space M_g of genus g Riemann surfaces. In recent years, McMullen has completely classified Teichmuller curves in M_2, but little is known in genus three and above. In this talk, I'll discuss the problem of classifying Teichmuller curves and progress (in joint work with Martin Moller) in the genus three case.
(March 3) Jayadev Athreya: Gaps in angles between generalized diagonals — Given a Euclidean polygon P, a generalized diagonal is a billiard trajectory connecting one corner to another. In joint work with Jon Chaika, we consider, for rational polygons P, the question of how close in angle two generalized angles of length at most R can be.
(March 17) Cheng Chong-Qing: Regularity of weak KAM and Arnold diffusion in a priori stable system with 3 degrees of freedom — In the contruction of diffusion orbits, one key point is to establish the transversality for the intersection of stable and unstable sets. It is closely related to the problem of regularity of weak KAM. Once the regulairty is obtained, one can construct diffusion orbits by the variational method based on Mather's theory. In this talk, I shall show how to do it for a class of a priori stable Hamiltonian systems with 3 degrees of freedom.
(March 31) Martin Schmoll: Dynamics on Panov planes — Recently several groups started to study the dynamics on Z-covers and Z^2-covers of (half)-translation surfaces. We recall selected results and some constructions particular for Z-covers. Then we go over to Z^2-covers and display recent results concerning the Ehrenfest windtree model, studied by Hubert, Lelievre and Troubetzkoy and Panov planes.
Panov planes are a generalization of the folded complex plane constructed by Panov relatively unnoticed about 10 years ago. Panov planes arise when flat pillow cases (topologically spheres) are attached to slits of a doubly periodic slitted complex plane. The result is topologically still a complex plane and carries a flat (more precisely: half translation) structure with cone singularities. The direction flow (along straight lines) for any direction on the complex plane is defined on Panov planes.
Panov's result describes folded planes having (a dense set of) directions containing dense orbits. We give a generalization of Panov's construction also carrying dense orbits and relate it to the Ehrenfest windtree model by a covering construction. That allows us to move and compare results for both models.
This is work in progress with Chris Johnson (graduate student)
(April 7) Lior Fishman: Schmidt's game, friendly measures and exceptional sets on fractals — In this talk I shall describe new results regarding properties of certain sets on fractals. Questions regarding these sets, often exceptional both measure and category wise, arising from number theory, dynamics and Diophantine approximation theory, have been extensively studied in recent years utilizing Schmidt's game and properties of the class of friendly measures.
In order to highlight the main ideas in many of these proofs, I shall (time permitting...), reprove a slight modification of Schmidt's original result regarding the set of badly approximable numbers, pointing out where generalizations have been made using modern ideas and techniques.
I wish to emphasize that the talk will be quite self contained, thus accessible to graduate students as well as anyone interested.
(April 21) Nelson Markley: Completing the Weil-Hedlund-Anosov Program — The talk will begin with a discussion of what I mean by the Weil-Hedlund-Anosov Program for studying flows on compact connected surfaces followed by examples of results that illustrate the natural interplay of dynamics and geometry in this program. The third part will be devoted to introducing a theorem that allows us to prove the two fundamental theorems of Maier in this context and bring the program closer to completion.
(May 5) Dmitri Scheglov:
Explicit splitting for triangle billiards —
For a class of irrational triangles we provide an explicit estimate on the
splitting time of a thin parallel beam of trajectories. The key argument of the
proof includes search for long monochromatic arithmetic progressions for some circle
pseudo- dynamics. Possible generalizations will be discussed.